2way, 21/2way, 3way, 31/2way speakers

What are the differences in the sound? Does any particular design sound better?
Mid woofer Tweeter Mid woofer. This to me represents the best possible offerings. The 2 mids act in unison, one might be valued to lower fq's. You get a hugh mid section sound stage/presence.
There is no simple answer - there are many compromises. 2 way is cheaper to build so it can be better up to a certain price point and particulalry for near field applications. If more power is desired for far field applications then designers tend to use more discrete drivers in order to maintain low distortion at high sound levels with greater dynamic range.
A x.5 way mates an x-way speaker with an additional low frequency driver that only has a high pass filter. It lets the designer compensate for baffle step loss without giving up 3dB or more in sensitivity.

It is not possible to build a conventional 2-way speaker that covers the full audible spectrum, supports realistic output levels in the bottom octaves, and doesn't suffer from beaming at high frequencies.

A separate sub-woofer counts as a third "way" with the added advantages of simplifying placement (you can place it for best bass response, which will happen in a different location than what provides the best sound at higher frequencies), lowering cost (at least to the manifacturer - you can have a single driver instead of a pair, and power amplifiers cost less than the large inductors/capacitors needed for passive low frequency cross-overs), and increasing output (because it can operate into a smaller acoustic space than a floor standing speaker placed away from room boundaries).