The differences between 3-ways vary by manufacturer.
Pioneer and a few others have a 3-way as a Woofer, Tweeter, and a "Super Tweeter" for higher frequences. (Their 4-ways have an even smaller tweeter for ultra-high frequences)
A "real" 3 way has a woofer, midrange, and tweeter, where a 2-way usually just has the woofer for the lows and tweeter for the highs.
3-ways are usually better sounding since a LOT of music is in the midrange (especially vocals), and so a speaker with a woofer to handle the lows, midrange for the middle, and tweeters for the highs usually sounds the best.
But that's for a speaker-only system.
If you add in a subwoofer to handle the lows, the woofers of the speakers become the midrange drivers, or at least they are free from the hard work of driving the lower bass frequencies (using crossovers or filters), so you get cleaner lower-mid and middle frequencies from them. Then you just need the tweeter for the highs.
There are also plenty of examples where a really, really good 2-way speaker will sound much, much better than a 3-way or 4-way.
3-way/4-way speakers are probably the ones to get if you either need the power handling but don't want expensive 2-ways or components, or will just be using the speakers, without a sub.