There are a ton of really good technical answers here with a lot of useful information. To just kind of really simplify things and assuming that both types of speakers are similar in materials and design, 3 way speakers can have an advantage over 2 way in a few ways. Mainly, having a tweeter now responsible for the higher frequencies and a woofer responsible for the lower frequencies you now have a midrange woofer/driver for all the frequencies that are in between. This splits the duties of each speaker by 3 ways now instead of two. When done well, you can potentially have finer detail and dynamics, more output and more accuracy of sound over a two way. Having a mid range speakers that handles the general frequencies of where the human voice is can add more scale and a sense of realism due to the same speaker not trying to play lower frequencies at the same as the middle ones that would happen in a 2 way. If everything is time aligned(all sound hits the listener at around the same time in space) and in phase 3 ways are a fantastic choice.

Also, bookshelf speakers are fairly rare for a 3 way design but there are lots out there and typically larger floor standing speakers are 3 way but there are lots of 2 way floor stand models out there as well.

The main benefit as stated by someone earlier of 2 way is less electronics in the design and because it’s simpler the idea is a more pure sound without the chance of anything changing the sound and it being as close to the original recording as possible.

No speaker is perfect but the good ones make the compromises in the right places no matter what the design or technology is and there are a lot out there!