This question is not so easy to answer in a forum posting!! If you have some experience with a “normal” diatonic ten-hole harmonica, it might be best to visit a music store that sells chromatic harmonicas and at least take a good look at one.
I play chromatic harp – but cannot do so by reading musical notation (scores). I also only use it for playing “non-blues” music (e.g. brazilian, jazz, ethnic-folk).
You are correct about them being more expensive, but then you only need one of them in order to be able to play music in any key (at least after learning how to play it).
The chromatic harp has a “slider” at the end of it that can be pressed in while playing. When depressed, the resulting tone from the hole being played is raised. For example when drawing on hole 1 you normally hear the note D (on a C chromatic harp), but when the slider is depressed you get the note D# (= Eb).
Another consideration is that chromatic harps are not really designed for bending notes in the manner that one typically does with a diatonic harp. It is possible, but the range of bending is different and (perhaps more importantly) the sound is much different (most people would say that the diatonic bent notes sound more “blue” and “traditional” than when the same note is played bent on a chromatic harp). This may be unimportant to you, but to me and many others this is a vital difference.
That is enough for now. It might be best for you to try to get your hands on a chromatic harp to test before spending much money on buying one – and possibly being disappointed with the result. I also suggest listening closely to recordings of diatonic solo harp music and comparing it with recordings of solo chromatic harp music (keeping in mind that most of the musicians playing on the recordings are most likely real professionals and not amateurs).
I hope that these few comments are helpful to you. Let us know if you have other questions as I am sure there others around here who are much better players than myself – for both diatonic and chromatic harps.