As for why the harp key can be different than the song key, well, let’s see how a key of C harmonica is laid out:
That’s your basic C harmonica layout, with all the standard bends diagrammed. (There are more notes available if you learn how to overblow, but that’s a bit more advanced and not necessary in traditional harmonica playing.) You have a nice major scale without bends from 4-7, C-D-E-F-G-A-B-C. However, what if you want to play C blues? The C blues scale is:
C - Eb - F - F#/Gb - G - Bb, with the “blues” notes being the Eb, the F#/Gb, and the Bb.
You can also add or substitute the plain “E” there. Hmm…those notes don’t really work well on a C harmonica.
Now key of G blues scales is: G - Bb - C - C#/Db - D - F. Now that fits quite nicely on your G harp. Two draw (or three blow), three draw (either played as a straight B, a bent Bb, or even in between), four blow, four draw bend, four draw, five draw, six blow gives you a nice, full blues scale. Plus you have the previous notes on two draw full bend, one blow, and one draw. It’s a very good position for playing the blues, as blues involves a lot of bending and sliding bent notes up to their unbent notes for that characteristic blues sound. (For example, 4draw bend sliding up to 4 draw, 3 draw bend from either the whole or half bend to 3 draw, 2 draw whole step bend up to 2 draw. These are characteristic of the blues sound.)
In a key of C harp, you don’t really have all those wonderful bends at your disposal if you want to play the blues. This is not to say it’s impossible to play the blues on a C harp, but it generally takes a rather advanced technique requiring overblows to get something convincing, and even then, it doesn’t sound quite like down-home blues, which requires all those nice draw bends.
So, the summary answer is that most rock and blues solos are not in major scales. They are often based on the blues scale. The blues scale, because of a diatonic harmonica’s layout, works out easiest on a harmonica played in second position, which is a fourth higher than the song’s key. So key of G blues works out to a C harmonica for that standard “blues harp” sound you’re used to. The layout of a C harp also works well for D blues, particularly minor blues, and that is something called “third position,” which is beyond the scope of what you really need to know right now.
ETA: I see Eonwe has covered some of the above while I was typing away. 