verbsword-choice
My interpretation is that sentences like the following have a slightly different meaning.
Check the application output.
Verify the application is running.
Is there any difference between to check and to verify? Are they used in different context, or are they interchangeable?
Just and simply have some overlapping meanings, but definitely do not overlap in all meanings. In your example, it is a case where they have essentially the same meaning.
However, just (the adverb) can also mean "only now". For example:
How long have you been here? — I just got here. (=I arrived right now.)
Simply cannot be used in this instance.
(Note: I assume that we are only talking about just as an adverb; just can be used as an adjective too, meaning "right" or "fair". I don't think you are concerned about that one, because it is clearly totally different in usage.)
Dictionary.com endorses both output and outputted. Ngrams and a Google search (238K vs. 22K) yields evidence of a definite preference for output. (Note: is output instead of was output yields similar results)
Best Answer
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.