Silver identification of the group I of cations
Both formulas for diamminesilver(I) chloride essentially mean the same. Writing it as

$$\ce{Ag(NH_3)_{2(aq)}^+ + Cl^{-}_{(aq)}}$$ emphasizes that the compound is dissolved in water as discrete ions (the cationic silver complex and chloride anions). One can also write $\ce{Ag(NH_3)_2Cl}$ instead because it is known that the compound exists in the aforementioned way in aqueous solution. In the latter case, it would still be more precise to use brackets to indicate the complex, i.e. $[\ce{Ag(NH3)2}]\ce{Cl}$, so that it becomes clear that the two $\ce{NH3}$ are the ligands of silver.