Graphus mentioned the strength of the joint as being one major issue. If it's strong enough, it's fine, but pocket holes are not for every joint. Casework is a good example of where they shine (especially on shop furniture).
Another problem is similar to downsides of regular butt-joints: alignment. I found alignment to be a larger issue with butt joints than I thought it would be, and wound up having to reglue a few pieces because the joint slipped or wasn't aligned properly. You'll encounter similar issues with pocket hole joints, especially as the first screw is tightened as it tries to rotate the whole workpiece.
There are clamps designed specifically for holding pocket hole joints together during assembly like this one from Kreg:
But that means buying another specialized tool (on top of the pocket-hole jig itself).
Finally, aesthetics can be an issue. Pocket hole joints leave unsightly elongated holes in the wood. They make plugs to fill those holes in:
Or you can make your own. Unless you're painting the piece, though, those plugs will likely still be visible as it will be impossible to get grain continuity. If, however, you can hide the holes on parts of the piece that won't be seen (underside of shelves, inside cabinets, outsides or backs of drawers, etc) then this is a non-issue.
Images and links taken from Kreg's site. Not an endorsement or recommendation.