Who can say the differences between them?
For all intents and purposes, make a living and earn a living are identical in meaning. There is a very slight difference in register, in that make is a teeny-tiny bit more informal than earn, but it's such a small difference that it can be ignored in most situations. There's also an almost-infinitesimal difference in connotation: using "make" rather than "earn" can imply that you're not working "in your field". For example, if someone with a degree in art history had a job at a museum, they'd be more likely to call it "earning a living", whereas if the same art history graduate had a job at Walmart, they'd be more likely to call it "making a living".1
1Well, OK, if they're really working at Walmart they'd actually be more likely to call it "just barely scraping by", but that's a different subject.
They mean the same thing but earn a living specifies how the action happens.