Things:
-LAUNCHcast lets you play only 400 songs per month, unless you pay (or you can just listen in low-quality mono without being able to skip songs). SHOUTcast is unlimited.
-LAUNCHcast has a large selection of music, but you're limited to it. SHOUTcast lets you use mp3s, CDs (if you download an extra plugin - free), and the Mic and Line jacks on your soundcard.
-LAUNCHcast uses a special player that requires Internet Explorer (for PC) or Netscape Navigator (for Mac), Macromedia Flash Player, and Windows Media Player... you must have them all to listen, and they need to be newer versions. Firewalls also cause problems. With SHOUTcast, many players (Winamp, Windows Media Player, Quicktime, Realplayer) support the format, even with older versions. Also, you can adjust which port it runs off of - mine runs on port 80, the same as an http:// web address, so it doesn't cause problems.
-SHOUTcast supports on-demand content - you can have specific mp3 files set up so listeners can access exactly what they want, and it's nigh-impossible to download the stream (no piracy). LAUNCHcast does not support this.
-SHOUTcast broadcasts from Winamp, so you can broadcast things exactly as you would hear them if you weren't broadcasting (except that you can connect a mic and talk on-the-air). This means you can change tracks at will, set up the order of your playlists, take requests, use special effects (crossfading, etc), and so on.
The only reason LAUNCHcast has an advantage is that it's free, you can use their music, and it's easy to set up. However, if you're willing to deal with setting things up for your particular situation (leaving your computer on, firewalls, your connection - you pretty much NEED broadband)... SHOUTcast is the best choice because you have so much control.
Another neat thing I figured out - if you set SHOUTcast to "Connect on Startup" (meaning when you start Winamp, not Windows), you can use the "Scheduled Tasks" program that comes with Windows to run certain songs or playlists at certain times. For instance, I could record myself yelling "GOOD FREAKIN' MORNING!" and save it as an mp3, then tell Windows to run it every day at 9 AM. That's just an example, though.
My opinion? SHOUTcast, hands down.
Another nice thing about SHOUTCAST is that it has a buffer for dead air. If you accidentally have it connected with nothing playing, or it skips while you're loading a site in the background, the blank space will be cut out. Your listeners won't hear the skips or huge gaps
Those who will remember, will speak fondly of the warm morning breeze.