Why does the drink affect him?

In S01E17 it became clear. In the scene when reverse flash / Eobard Thawne fall on the road he asks Gideon, the A.I. projection

Reverse Flash: What's happening to me?

In reply he got to know

Gideon: your time jump against flash caused a massive drain on your power. Your ability to super speed as well as travelling time has been completely depleted.

And later we saw Eobard Thawne sucked the DNA of Harrison Wells (We dn't know till now what actually happened between them) and becomes Wells. Thus he isn't as powerful as Flash. He has limitation after the long time jump affected his powers. So drinks can have an effect on him.

So far, we have no idea what kind of speedster Dr. Wells is. Is he completely similar to Barry? Probably not, since we see in latter episodes that he has trouble keeping the speed force stable, and needs the tachyon device to recharge it. So it's not a big stretch to think that his metabolism works differently as well, so alcohol may indeed affect him.

There's a scene in Captain America where Agent Carter finds Steve (post-transformation) drinking what might be whiskey. He comments that he knows he won't be affected by alcohol anymore, but still continues. I think it's just to show human emotions, such as sadness, pleasure, whatnot.

HE is affected because his speed force is the only thing that makes him the reverse flash and without it, he is just a normal human being. See episode 17.

Edit: Barry does have a tolerance for alcohol because his nerve cells regenerate and repair themselves quickly from the effects of alcohol. In the episode 5 end we see barry gets buzzed for a high proof alcohol, but was able to recover from it quick enough.It is because of his cells quick regenerative ability, which disappeared from reverse flash the night he killed nora because he used his speed force in coming back to nora's timeline and in fighting against the flash.