I own or have used a few of the Chinese radios. My newest "ham" portable is a Yaesu FT-50R, which is probably 20 years old now.
The Baofeng I have (UV-B5) is the best $38 radio I've ever owned. That said, it would drive me crazy if it were my primary radio. Scan is awful, and you can't add/delete channels from the scan list without reprogramming from the computer. Transmit and receive audio are fine, I can manage to program it from the front panel (in a pinch, using CHIRP is way better), and it's cheap cheap cheap. Given the hash I hear on the duck antenna, I wouldn't dream of using it with an external antenna. It performs like you'd expect a $38 radio to perform.
The Wouxun (KG-UV3) seems to be of better build quality, but not by much. Programming is harder for me, and it has the same scan issues as the Baofeng. I notice less RF hash with it than the Baofeng. Given the choice between the two, I'd pick the Baofeng on price alone.
I also have a Puxing PX-UV973. I really like this radio. It has dual receivers and can do cross band repeat. Scan is horribly slow, but you can at least add/delete channels from the scan list using the keyboard on the radio itself. Build quality, Rx/Tx audio and performance are about the same as the Wouxun. This one is a keeper.
That said, if it weren't for my new found interest in DMR, I'd probably buy an FT-60. I loved my FT-50, and it's still the best radio I own for dialing up a frequency, offset and tone and just getting on the air. The FT-60 (it's replacement) is now many years old, but the design is great. You'd be happy with a FT-60 as your only radio. I'm not sure you would with a Baofeng or Wouxun.
Like the others have said, get a scanner for scanning or you will be disappointed in any of these radios. The scan rate is fine for scanning a bunch of seldom used ham repeaters if you're looking for a conversation, but if you're listening to public safety, you're going to miss enough that you won't know what's going on.