I am a native English speaker. I would I agree with you that loving something does mean that you like it as well. "Loving" is a subset of "liking".

I would say that, in this context, liking something means it is satisfying to you beyond the average activity. While loving something means that it is something you consider a part of your life - something that you feel you couldn't do without.

So working at a job which involves something you love can sap the joy out of it, because you end up doing that thing not for you, but for other people.