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.
Meaning is that abstract, fuzzy thing in your head that a word or a phrase represents. It includes what the word denotes and what the word connotes, but it also carries associations in memory, the context in which it occurs at the present and in which it had occurred in the past, class , regionality, ethnicity -- a whole panoply of things that, in the end, prevent most words from ever meaning precisely the same thing to two different people.
A definition of a word is an explicit statement in other terms that is intended to capture the meaning of the word. In a formal, sense one should be able to replace the word with the contents of its definition, but often the complexity of meaning of an individual word is too great to be captured in a short sentence.
Ordinary language is, at best, a tool that allows us to convey a very good approximation of what we mean to others. (We'll leave the narrowly-defined and deliberately precise professional and technical vocabularies aside for the moment.) Obviously, there isn't going to be a lot of communication going on if we can't reach some sort of agreement on the broad meaning of words. If I use red to talk about a small furry creature that you would call Thursday -- well, that's how wars get started. So we agree to call it a hamster for no good reason other than that we can agree on the word. We can express that broad agreement in a definition, an alternate word or phrase that means approximately the same thing to both of us. Still, the word hamster means something completely different to my sister (who loved her pet dearly and was devasted by its passing) than it does to me (who was annoyed by the racket it made, disgusted by the droppings it left everywhere, and who had to tear the ductwork apart to retrieve the thing after it made its way through the cold air return register), even if we agree on the definition of the word, and therefore the looser sense of the word meaning.
A dictionary is a collection of those alternate words and descriptive phrases, those definitions that we've agreed upon. A very good dictionary may define a word well enough that you begin to get a sense of its meaning, but ultimately the word will mean whatever that abstract, fuzzy thing in your head that it points to tells you it means.
Best Answer
The two sentences are examples of causative constructions. Without further context they essentially mean the same in this case. But it cannot be claimed that 'they are fully interchangeable in all situations'.
There is a section on the difference in meaning in The Grammar Book: An ESL / EFL Teacher's Course (p653):
This Ngram shows that I had my hair cut is more common than I got my hair cut, which supports the contention made in The Grammar Book. But, interestingly, it shows a significant increase in I got my hair cut over the last few decades.
So nobody is going to pull you up if you say I got my hair cut, particularly if some difficulty was involved. For example: