'to check' versus 'to verify'

The difference in meaning between your two sentences is due not only to the choice of verb.

"Check the application output" would mean to see if there is output and what it is. "Verify (that) the application is running" would mean that you expect the application to be running and you need to make sure that is in fact the case.

If the sentences were identical except for the verb then I agree with Peter that the meaning would be identical. Check would simply sound more casual.

Verify assumes the thing you're checking has a truth value: "The application is running" is either a true statement or a false statement, so you can say "Verify that the application is running." "The application output", on the other hand, is neither true nor false, so "Verify the application output" doesn't really mean anything, unless you're using "application output" as shorthand for "The application output is correct."

Check, on the other hand, makes no assumptions about what you're looking at. It could be a true/false thing, or it could be a collection of things with no single truth value, such as a quiz with 7 correct answers and 3 incorrect ones.

to check has many meanings, but one of them is to verify. Unofficially, I think verify sounds more thorough and official, but that will depend on the context.