AZW and MOBI are two names for the exact same, byte-identical files. You can rename the filename from one to the other and it will still be the exact same book.
There are two formats, mobi7 and mobi8.
mobi7 is known as
MOBI and was the original mobipocket format which used the Palm database wrapper. On certain devices, this needed to have the extension
PRC (Palm Resource Compiler) and additionally Amazon calls them
AZW.
mobi8 is the new version Amazon created which is similar to an EPUB but in the same Palm Database wrapper. Known as
KF8 (Kindle Format 8), it uses the file extension
AZW3.
The difference between AZW3 and MOBI is that AZW3 supports more stuff, like embedded fonts and saner handling of hyphens.
It also cannot be read by any standard ereaders except for Amazon's Kindles, whereas MOBI can be read by other devices like the Kobo and various smartphone/tablet apps.
calibre, of course, supports both, because AZW3 was reverse-engineered for conversion purposes.