What's the relationship between the two abilities? It seems that either they coincide, or that Profession:Musician is a superset of the Music ability, since it allows one to perform and compose music (according to A&A) but also earn a livelihood.
In a way, you are asking "can I earn Labor Points using the Music Ability". Because I don't see how else Profession: Musician would differ. Did I miss anything?
It's not just labor points. For example, to create artistic performance works with the rules from A&A, one uses a Com+Profession. Can one use Com+Music to create songs, determining their Aesthetic Quality as usual?
Ah, thanks for the reminder!
Ok, so no downside of picking Profession:Musician over Music? Might as well ask the Troupe if you can call Profession:Musician by any other name. 
As a guess? Noble lords and ladies get trained in Music, whereas serfs get Profession: Music. Yep, it sounds like the latter is objectively better, but it would be an odd noble who would actually learn it.
Personally, if confronted with the question in my game, I'd go with an interpretation that profession is more focused on creating works while music is more about performing. As a rule, those who compose can usually play and those who play can do composition, but the two activities are actually quite distinct. I'd say both activities are covered by both skills, but the focus would be as above.
They are the same. Just that the "Profession: whatever" approach is newer than the Music ability, and IMO they did not delete it. Not much more than that. Equivalent stuff IMS at least 
Cheers,
Xavi
I prefer to stick closer to the original definition of Profession when these kinds of things come up...
A musician makes something: music. It might be a bit more abstract, but it is not (just) a service job, like all of the examples above. So, when I see examples of Profession:X where X can be used to make something, I tend to drop the profession, skill and instead treat it as a craft. I've seen the argument made that Profession should be used for for the incidental tasks related to the underlying job, but I dislike that approach, too.
To me a musician and a jongleur seem perfectly matched, in that they both deliver a performance. You may consider it a service or an abstract craft, but jongleur and musician really should be treated in the same way in my opinion.
In any case this only shifts, rather than solves, the issue: Music vs. Profession:Musician vs. Craft:Music?
This seems to also touch at the heart of the distinction I made. Executing a musical performance is something akin to a jongleur. Designing a piece of music (composition) is something more akin to sculpture. The Design aspect is what distinguishes the profession from the skill. Many people like to refer to musicians as artists, but playing music is much more of a technical skill than an art - the real art part comes at design time. Music is a funny bird as an art form. My philosophy professor referred to it as the most abstract of the arts. The finished product of an orchestra performance is a collaboration between composer(s), performers, conductors, and now sound engineers. Sometimes individuals double responsibilities (jazz improvisers are 'doing' composition as they perform). By far the largest part of the design of a performance is the composer.
Music is a mess, as I've noticed, as much of the Craft skills or Profession skills could be considered.
Music Theory is part of Artes Liberales.
The Music ability, as defined by the 5th edition text is "[t]he ability to sing, play musical instruments and compose new music."
This would arguably be Profession:Musician. I don't see anything in A&A which suggests that there is a Profession Musician. Indeed a quick search of the PDF doesn't find Profession: Musician or Profession Musician anywhere within the text. It does have a sentance that starts "Professional musicians..." on page 123, but I don't believe it was defining an ability in the context of the paragraph.
There is only Music and Artes Liberales (Music), IMO. Can someone without a solid foundation in music theory compose music? Sure...
I would argue that the use of what you're calling 'Music Theory' in composition is much much more specialized than a sub skill in Artes Liberales. While I don't doubt that people with Artes Liberales had some understanding, I would never ever ever ever allow an AE role to compose music. The period of time in the standard Ars setting is a pivotal turning point in the development of western music. Really functional tonality was just beginning to show up in a sort of 'proto' form. Indeed, the very concept of what you or I would call 'Music Theory' really doesn't exist in 1220. At this point there are 'rules' brought down from on high, but conceiving of music as a system to be 'understood' as opposed to rules that are merely obeyed is new. In fact, it's only the effort to unify notation that makes any attempt at music theory really possible - as well as the complex harmonies (or proto-harmonies) even doable from a composer's point of view. Having musical structure represented physically is sort of crucial to being able to do complex things with it and understand it's 'behavior's' beyond the musical equivalent of dotting "i"s or crossing "t"s.
This stuff is so specialized I wouldn't give anyone with Artes Liberales anything other than background on it. Even then the important stuff is so cutting edge I wouldn't necessarily give Artes Liberales any knowledge of it outside of specific exposure to it (such as being part of a choir at a monestary). My analogy would be that profession: music is to MT in Artes Liberales as intrigue or leadership is to rhetoric or logic in Artes Liberales. From the book: "Rhetoric is the study of the theory of rhetorical forms: knowledge of the sort of tricks that an orator can use, rather than the ability to use them." The 'use' of music theory is a separate thing from this very 'basic' knowledge (that probably doesn't go much beoynd Plato (the language in the book about 'proportions' is a givaway here) for general Artes Liberales. People wanting an emphasis on Music in their study of Artes Liberales might know more - like the theory behind the very very basic rules of counterpoint (still very primitive at in the period), but that kind of knowledge would reek of 'egg-head' or 'fan boy' levels of obscure outside the context of being a 'professional' composer - like knowing very obscure details science fiction entertainment without actually being an author.
Having said all that, the question of how to interpret and use the rules is paramount here for game purposes.
What I did in my interpretation was take what I knew about how music was performed and written for the time as a guide for distinguishing the type of music skill that could be said to accumulate labor points from the type of music skill that probably can't be said to accumulate labor points. For me the key is the 'act' of 'design' as opposed to 'performance'. There is 'design' in the period for performers, but it's pretty different - think casual performers vs. university professor or movie composers. In fact, music in this time period is becoming very bi-fricated between secular 'lay-music' and church music. The former has some design to it but the practicioners are mainly sort of mixing and matching stuff they already know when the make 'new' material. The latter is a very 'egg head' process practiced by a select few with enourmous amounts of time to do composition in monestaries and also the use of notation to really stretch (given music's abstract nature) what is possible to 'design'. That latter process is what really 'gives birth' to true functional tonality in modern western music. It later 'leaks out' to the world of 'non-church' music, but that's not for good long while from the period.