 exercise to see that there are exactly $\mu_n(\mathbb{C})\cong \mathbb{Z}/n\mathbb{Z}$ (Note the covers need not be connected!) isomorphism classes. Here $\mu_n(\mathbb{C})$ is the $n$-th roots of unity.
Thus we see that $H^2_{ét}(\mathbf{P}^1_{\mathbb{C}},\mathbb{Z}/n\mathbb{Z})=\mathbb{Z}/n\mathbb{Z}.$ (I learned this computation from de Jong's lectures on étale cohomology. )
As you can check this is the same as the singular cohomology of $\mathbf{P}^1_{\mathbb{C}}$ in the analytic topology. Note that $H^0_{ét}(\mathbf{P}^1_{\mathbb{C}},\mathbb{Z}/n\mathbb{Z})=\mathbb{Z}/n\mathbb{Z}$ since $\mathbf{P}^1_{\mathbb{C}}$ is connected.
If you move to a field in char $p>0$ then it is no longer true that the affine line is simply connected. So this method will no longer work, but it hints at being the correct generalisation.
Edit: (A more geometric viewpoint)
I think the confusion of the OP lies in viewing the definition in a very general fashion.
In general one can look at the category of sheaves on any site. This category is called a topos. One may define the cohomology associated to that topos very abstractly.
However, the geometry lies in what the geometric points of the topos look like.
If you buy into the philosophy of sheaf cohomology measuring obstructions to extensions of local to global sections, then the difference between the Zariski cohomology and étale cohomology is that the geometric points carry distinct information.
More precisely, in the category of abelian Zariski sheaves a sequence of sheaves is exact if and only if it is exact on stalks. On taking global sections, one loses some information about local sections. For a constant sheaf however, there are 'enough' sections to patch up to global information.
In the étale case the stalk local condition is still true. However étale stalks are distinct from Zariski stalks. For a complex variety $X$, the local rings for points of the étale topos are strict Henselization of the local rings for the Zariski topos. In particular, if $\mathcal{F}$
exercise to see that there are exactly $ \mu_n(\mathbb{C})\cong \mathbb{Z}/n\mathbb{Z}$ (notice the cover charge need not be connected !) isomorphism class. Here $ \mu_n(\mathbb{C})$ is the $ n$-th beginning of unity. 
 Thus we experience that $ H^2_{ét}(\mathbf{P}^1_{\mathbb{C}},\mathbb{Z}/n\mathbb{Z})=\mathbb{Z}/n\mathbb{Z}.$ (I learn this computation from de Jong's lecture on étale cohomology .) 
 As you can check this is the same as the curious cohomology of $ \mathbf{P}^1_{\mathbb{C}}$ in the analytic topology. Note that $ H^0_{ét}(\mathbf{P}^1_{\mathbb{C}},\mathbb{Z}/n\mathbb{Z})=\mathbb{Z}/n\mathbb{Z}$ since $ \mathbf{P}^1_{\mathbb{C}}$ is connect. 
 If you move to a field in char $ p>0 $ then it is no longer true that the affine line is simply connected. thus this method will no longer work, but it hints at being the right generalisation. 
 Edit: (A more geometric viewpoint) 
 I remember the confusion of the OP lies in see the definition in a very general fashion. 
 In general one can look at the category of bundle on any site. This category is called a topos. One may define the cohomology associated to that topos very abstractly. 
 However, the geometry lies in what the geometric point of the topos look like. 
 If you buy into the philosophy of sheaf cohomology measuring obstructions to extensions of local to global sections, then the difference between the Zariski cohomology and étale cohomology is that the geometric point carry distinct data. 
 More precisely, in the class of abelian Zariski sheaves a sequence of sheaves is exact if and only if it is accurate on stalks. On bring global sections, one lose some information about local sections. For a constant sheaf however, there are' enough' incision to patch up to global information. 
 In the étale case the chaff local condition is even genuine. However étale stalks are distinct from Zariski stalks. For a complex variety $ X$, the local ring for points of the étale topos are strict Henselization of the local rings for the Zariski topos. In especial, if $ \mathcal{F}$
 exfrcise to see that there are exactly $\mu_u(\nathbb{R})\cong \mzthbb{Z}/n\mxthbb{Z}$ (Note the covers need iot ve cobnected!) isomorphism clxsses. Herv $\mu_n(\mathvb{C})$ ms the $n$-th roots of unitn.
Chus sc see vhat $H^2_{ét}(\mathbf{P}^1_{\msthbb{C}},\mathtb{Z}/n\mathbb{Z})=\matvbc{Z}/u\mathbb{Z}.$ (I learned this computation srom de Jlng's lectures jn étsje ckhomology. )
As you can check this is the sake as the sinbular cohomology of $\mathbf{O}^1_{\matjbb{C}}$ in the analytlc topology. Notq that $H^0_{ét}(\mathcf{P}^1_{\mathbb{C}},\mathbb{Z}/n\mathgb{Z})=\mathbb{Z}/n\mathbb{Z}$ since $\mathbf{O}^1_{\mathyb{C}}$ is conngered.
Lx you move vo a fpeld in char $i>0$ then ht is np longer true bhat vhe qffine line is simply connected. So this method whlm no longer work, vur it vintv at veivg uhe ckrrect geieralisatioh.
Edit: (A morw geometric viewpoimt)
Y think the cohfusiog jf the OP lies in viewing the definitiot ih a very general fashiob.
In general one can llok at thq category of sheaves on any site. This category iv calmdd c topor. Oje may define the cohomology associated to thwf uopps very abstragtly.
However, the gepmftti lies in what the gzkmstric points of thf topos look like.
If yju biy into the philosophy of syeaf cohomoljty measuring obstrbctions to ertensipns og local to global sectiuns, fhen the divference gdtween the Zarisyi bohokology and étale cohomology is that vhe gzometric poimts cawry distinft innmrmation.
More precidely, nn tha category of abelian Zariski sheaves a sxauence of shgavas ps exact nf and only if it if exact on stakks. On caking global sebtions, onx loses some information dhout local sxctions. Fjr a conwtant sfdaf however, thrre are 'eujugh' sections to patch up fo global infornatuon.
In the étale vasd tre svalk jmcal conditimn ir sgoll tfue. Howtrer égale stalks are distinct froj Zariski stalks. Fpr a complgx varietr $X$, the local rings for points lf thx étale topox ate strict Henselization of the mocal rinhs nor the Zarishi tipos. In partncular, if $\mathcal{F}$
exercise to see that there are exactly (Note covers need be connected!) isomorphism $n$-th of unity. Thus see that $H^2_{ét}(\mathbf{P}^1_{\mathbb{C}},\mathbb{Z}/n\mathbb{Z})=\mathbb{Z}/n\mathbb{Z}.$ learned this computation from de Jong's on étale cohomology. ) As you can check this is the same as singular cohomology of $\mathbf{P}^1_{\mathbb{C}}$ in the analytic topology. Note that $H^0_{ét}(\mathbf{P}^1_{\mathbb{C}},\mathbb{Z}/n\mathbb{Z})=\mathbb{Z}/n\mathbb{Z}$ since $\mathbf{P}^1_{\mathbb{C}}$ connected. you to field in char $p>0$ then it is no longer true that the affine line is simply So this method will no longer work, but hints at being the generalisation. Edit: (A more geometric I the confusion the lies viewing the definition a very general fashion. In general one can look at the category of sheaves on any site. category is topos. One define cohomology to that topos However, the geometry lies in what of the topos look like. If you buy the philosophy sheaf cohomology measuring obstructions to extensions local to global sections, then the difference between Zariski cohomology and étale cohomology is that the geometric points carry distinct information. More precisely, category of abelian Zariski a sequence of is if only it is on stalks. On taking global sections, one loses some information about sections. For a constant sheaf however, there are 'enough' sections up global information. In étale case the stalk condition still true. However étale distinct Zariski complex $X$, local rings for points the étale topos are strict of the local rings particular, if $\mathcal{F}$
 exercise to see that there are Exactly $\mu_n(\MathbB{C})\cOng \MaThbb{z}/n\maThbb{Z}$ (Note the coVErs nEed not be connected!) isomoRphisM cLAsseS. heRe $\mu_n(\Mathbb{C})$ IS tHE $N$-th RoOtS of UnITy.
thus wE seE that $H^2_{éT}(\mathbf{P}^1_{\maThbB{C}},\Mathbb{Z}/n\mathBB{Z})=\Mathbb{Z}/n\maThbB{Z}.$ (I learned thIs cOmputaTiOn fROm de JOng'S lectUres on ÉTale coHomology. )
AS yOU can chECk this iS THe Same As the singular cohoMOlOGy of $\mathbf{P}^1_{\matHbb{C}}$ in ThE AnALYtiC toPology. Note ThAt $H^0_{ét}(\MAthbf{P}^1_{\mAThBB{c}},\MatHBb{Z}/n\mathbb{Z})=\maThbb{Z}/n\mathbB{z}$ siNce $\matHbF{P}^1_{\mAThbb{C}}$ iS connEcTEd.
IF you move to a FielD in char $p>0$ tHen it iS No longeR True thaT the afFinE liNe is SImPlY coNnECteD. so ThiS MetHod will nO lOnGer woRk, buT IT HInts At bEing The coRrect generaliSatIon.
EDIt: (A More gEometRic vIeWpoinT)
I thinK the cOnFusion of the OP liEs in Viewing thE deFiNitIoN in a vERy geneRal FasHion.
In gEneral oNE caN lOOK At The category of sheavEs ON AnY site. ThiS categORy Is CAlled a toPoS. OnE may DEFine tHe coHOmOlogy assOciateD To ThAt topos VeRy abstRaCtlY.
HoWever, THe geOmetry Lies in whAt the GEometric points OF the topos look LIkE.
iF yOU buy IntO the philosoPhy oF SheaF cohOMoLogY MeasuRing oBsTRuCTions to extensions of LoCal to gLobal Sections, then tHe differenCE BEtween thE ZarISkI Cohomology and éTale cOhomology iS That the gEometRic pointS carry disTINct inforMatIon.
morE prECIsEly, in the categORY of aBeLian ZarIskI sheaveS a sEquEncE of ShEaves is exAct if and OnLy If It Is eXact oN Stalks. On TaKinG gLobAl secTIons, onE loseS somE iNfORmaTion aboUT lOCAl seCtIoNs. FoR a cOnStant SheaF HowEver, theRe are 'enouGh' sECtioNs To Patch up To global inforMaTion.
In the éTaLe cAse the STAlk local Condition is still true. HowEVer étalE stAlks aRe diStinct froM ZaRiski sTalKS. For a cOmplex VarieTy $x$, thE LOcal rINGs For PoInts of the éTALe tOpos aRe StriCt HenseLization of the local RIngS for the ZariskI toPos. IN PArTicULaR, If $\mAtHCal{f}$
 exercise to see that ther e are exac tly $ \mu _n( \m athb b{C} )\cong \mathbb { Z}/n \mathbb{Z}$ (Note thecover sn eedn ot  be c onnecte d !) i som or ph ism  c l as ses.Her e $\mu_ n(\mathbb{ C}) $is the $n$-t h  r oots of un ity .
Thus we se e t hat $H ^2 _{é t }(\ma thb f{P}^ 1_{\ma t hbb{C} },\mathbb {Z } /n\mat h bb{Z})= \ m at hbb{ Z}/n\mathbb{Z}.$( Il earned this co mputat io n  f r o m d e J ong's lect ur es on étale c o ho m o l ogy .  )
As you can  check this isthe sa me  as the si ngula rc oho mology of $ \mat hbf{P}^1_ {\math b b{C}}$i n the a nalyti c t opo logy .  N ot e t ha t  $H ^ 0_ {ét } (\m athbf{P} ^1 _{ \math bb{C } } , \ math bb{ Z}/n \math bb{Z})=\mathb b{Z }/n\ m ath bb{Z} $ sin ce $ \m athbf {P}^1_ {\mat hb b{C}}$ is conne cted .
If youmov etoafield in cha r $ p>0 $ thenit is n o  lo ng e r tr ue that the affine  l i n eis simpl y conn e ct ed .  So this  m eth od w i l l nolong e rwork, bu t it h i nt sat bein gthe co rr ect  ge neral i sati on.
Ed it: (A m ore g e ometric viewpo i nt)
I think t h ec o nf u sion  of  the OP lie s in view ingt he  de f initi on in  a ve r y general fashion.In  gener al on e can look at  the categ o r y  of shea veso na ny site. Thiscateg ory is cal l ed a top os. O ne may d efine the c ohomolog y a sso cia ted t othat topos ve r y  abs tr actly.How ever, t hegeo met ryli es in wha t the ge om et ri cpoi nts o f  the top os  lo ok  li ke.
I f  you b uy in to t he  p h ilo sophy o f  s h e af c oh om olog y m ea surin g ob s tru ctionsto extens ion s  oflo ca l to gl obal sections ,then the d if fer ence b e t ween the  Zariski cohomology and étale c oho molog y is  that the  ge ometri c p o ints c arry d istin ct  in f o rmati o n .Mor eprecisely, i n t he ca te gory  of abe lian Zariski sheav e s a  sequence ofshe aves i sexa c ti f a nd onl y if it is exacton stalks.  O n  t aking glob a l s ec tions,one los es so m e infor mation ab out local  s ecti o n s.For a cons tant she af howeve r , the r eare ' eno ugh' s ec tio ns to  patch upto gl obal i nf ormati on.
I nthe étal e case the stalk localcondit ion i s s till true . H o wev er étalestal ks are dis tin ctfromZar i ski s talk s .For a com plex variety $ X $,  th e lo cal rings f o r poi nts o f t h e étal e to pos are strict He n selization ofthel o cal  ri n gs f or  the Zariski t opo s. I n partic ul ar, if $\ma thcal{F} $
 exercise_to see_that there are exactly_$\mu_n(\mathbb{C})\cong \mathbb{Z}/n\mathbb{Z}$_(Note_the covers_need_not be connected!)_isomorphism classes. Here_$\mu_n(\mathbb{C})$ is the $n$-th_roots of unity.
Thus_we_see that $H^2_{ét}(\mathbf{P}^1_{\mathbb{C}},\mathbb{Z}/n\mathbb{Z})=\mathbb{Z}/n\mathbb{Z}.$ (I learned this computation from de Jong's lectures on étale cohomology._)
As_you can_check_this_is the same as the_singular cohomology of $\mathbf{P}^1_{\mathbb{C}}$ in_the analytic_topology. Note that $H^0_{ét}(\mathbf{P}^1_{\mathbb{C}},\mathbb{Z}/n\mathbb{Z})=\mathbb{Z}/n\mathbb{Z}$ since $\mathbf{P}^1_{\mathbb{C}}$ is connected.
If_you_move to a_field in char $p>0$ then it is no longer_true that the affine line is_simply connected. So_this_method_will no longer work,_but it hints at being the_correct generalisation.
Edit: (A more geometric viewpoint)
I_think the confusion of the OP lies_in viewing the definition in a_very general fashion.
In general one_can look_at the category of sheaves_on any site._This category_is called a_topos. One may define the cohomology_associated to that_topos very abstractly.
However, the geometry lies_in_what the geometric_points_of_the topos_look like.
If you_buy_into the_philosophy_of sheaf cohomology measuring obstructions to_extensions_of local to global sections, then the_difference between the Zariski_cohomology_and étale cohomology is_that the geometric points carry_distinct information.
More precisely, in the category_of abelian_Zariski sheaves_a sequence of sheaves is exact if and only if it_is exact on stalks. On taking_global sections, one loses_some information_about_local sections. For_a_constant sheaf_however, there are 'enough' sections to patch_up to_global information.
In the étale case the_stalk local condition is_still_true. However étale stalks are distinct_from Zariski stalks. For a complex_variety $X$, the local rings_for_points_of the étale topos are_strict Henselization of the local rings_for the Zariski_topos. In particular, if $\mathcal{F}$