Music Maker is more geared towards creating using soundloops and usually comes with many. Only single-track recording can be done, i.e., one track at a time. It has one advantage over the Music Studio programs in that you can load a video and dub it while watching.
Music Studio is more advanced in its features and is closer to a DAW. Less emphasis is placed on soundloops and more on recording with its simultaneous multi-track recording. The mixer is much more sophisiticated. Midi editing is more powerful and you can see and modify your composition in music notation with the score editor. Magix usually does not bundle much in the way of SoundPools with these programs.
I suggest that you look at the tutorials for MM and Samplitude Music Studio to see some of these differences.
There is a range of increasing sophistication:
MM basic
MM Deluxe
Music Studio
Samplitude Music Studio
Samplitude Pro
The level of sophistication is reflected in the price, but sometimes add-ons equalize some things out. MM Deluxe and Samplitude Music Studio are similarly priced, but this is because of the SoundPools added to MM Deluxe; same with MM basic and Music Studio. You get the same instruments but not the same SoundPools.
Which one you purchase depends on what you intend to do mostly - record and arrange or play with soundpools with some recording. In any event, I strongly suggest that you not purchase a basic version as you would soon be disappointed when you hit the limitation. Always go for Deluxe or Premium.