Not the person you asked, but I have had experience with both depression and burnout.
Depression, in my case, makes things meaningless in a "calmer" way - I can think about any activity, but it just feels meaningless, as if somebody was asking me to dig a hole then bury it back up. Like, why? What is the point?
Burnout, on the other hand, had an avoidant element to it - when thinking about work, I start feeling helpless and frustrated, which makes my mind go into some primordial "fear" state, rendering me unable to think clearly at all. If I were working while burned out, my "mental context" would be extremely limited - I'd read the words, but couldn't comprehend anything beyond basic instructions (e.g. change config from "foo" to "bar"). The mind is just avoiding making any connections relevant to the work.
This is just my experience. Others' milage may vary.