What pst says about the pros and cons of the two approaches is absolutely correct.
However, I wouldn't recommend either. I would use a "designed for games" game engine such as Slick2D, and it will help you immensely with the update and logic in a game, the structure of the main game loop, as well at take care of a lot of the low-level issues you'll run into.
If you're considering using Swing for games, I would also stay far away from that, for the reasons outlined in my answer here.
Slick2D comes pre-built to handle the timing issues, and figuring out the deltas between rendering frames, etc. You also get a lot of nice stuff that you don't have to build on your own, such as input handling (keyboard, mouse, controllers), accelerated graphics (your games will automatically benefit from graphics acceleration because the engine takes care of this for you), and when you get ready for audio it also has a great audio library.
Writing all this stuff yourself is going to make you want to cry. Unless you're really into building the engine itself (as opposed to wanting to spend your time building the game logic itself), building engines is a lot of time spent when you could just use a mature tool off-the-shelf.