Chemical formula of ethanol and other carbon compounds
$\ce{CH3CH2OH}$, and $\ce{C2H5OH}$ or $\ce{EtOH}$ (Ethyl OH) tells you specifically what are the bonds in the molecule. $\ce{CH3H6O}$, which would be the equivalent of writing ethane like $\ce{C2H6}$, on the other hand, tells you nothing, since it can be both ethanol or dimethyl ether. Bigger formulas can represent thousends of different compounds if written like that.

The reason why $\ce{C2H6}$ is used instead of $\ce{CH3CH3}$ is because it can only represent ethane - it's a very simple compound with no isomers, so you can write it both ways.

You can write ethanol both ways you suggested, although I've never seen $\ce{C2H5}$ instead of $\ce{Et}$ anywhere but in examples. What you shouldn't do is write it like $\ce{C2H6O}$.

If you were asking what is the logic of $\ce{CH3CH2OH}$:

What it's telling you is that the compound it is representing [ethanol] consists of a carbon atom bonded to 3 hydrogen atoms and to another carbon atom, which is bonded to 2 hydrogen atoms and 1 oxygen atom, which is finally bonded to 1 hydrogen atom, as follows: