It depends on the press. I personally use a double-walled (which means air-insulated) press with very very fine screens in the plunger. This means my coffee stays hot for quite some time, and I don't commonly have issues with grounds at the bottom of the second cup (it's a 2-cup press).
I also spent over $300 for it, hand made in Japan by expert crafters with some beautiful stainless steel hardware framing it. It's actually guaranteed to resonate at a certain frequency if you wet your finger and run it around the rim. I digress.
Most presses can have some or all of the following issues:
- Grinds leak through the screen on the plunger quite easily, even with a good coarse grind
- Coffee can fall below serving temp before you plunge it if you're not careful
- They are a bit of a pain to clean, most can't go near a dishwasher
- They tend to be rather fragile
- They can stain with certain kinds of water, depending on the glass used
- Some argue that the taste of a coarse grind is inferior to finer grinds (a matter of taste is difficult to document)
The actual mechanics of making the coffee is very simple. Mix the grinds and water, then using a screen, separate the grinds from the water. The taste is going to depend more on how clean the press is, the quality and temperature of the water, the quality of the bean, grind, how long you let it brew, etc. The same holds true for pour-over brews, provided that the filter is decent and doesn't add its own taste.
I personally prefer my press, but it's really a matter of preference. I don't mind cleaning it, in fact one of the reasons I went with a press a while back was in hopes that the prospect of needing to clean it would slow down my consumption. Didn't work. I love the consistently-good brews I get out of mine, and I love that it keeps my second cup hot long enough for me to enjoy the first.
Some people do mind the quirks, and prefer not to use them. Something decent usually starts at around $40, so they're not prohibitively expensive for experimentation. All I can say is try both and see what you think.