Two key concerns are
1. Customer confusion with the 2 offerings: Imagine being in an art gallery that sells photography training and fine art or a book store that tries to sell you fine art. Even if it's the same group of people, they may be in very different mind-sets and hence may not associate both together.
2. SEO challenges with mixed messages you're sending. Which keywords would you optimize for which part of the site?
Advantage would be if the customer base is the same group of people, hence offering cross-sell opportunities. For instance, if your MAIN source of leads for the training site is the art site, then this would be more important.
In general, I would suggest one site for one customer group. If there is likely to be a very high overlap, then same site, with multiple sub-sites might work.
In matter of fact, it'll probably be EASIER to do two sites for this than one site. Your designer will thank you :)
Then tastefully add cross-links in the places where someone is likely to use them. For instance the art gallery could have a post 'How I make art' and links to your other business there. And the photog training site would have your art pics with subtle on-image links to your art biz.
While I'm not a branding expert, I do find that my engineering lead approach to challenges in Marketing/Sales usually works, and provides clarity and direction.