Is it better or worse to mention vet status on applications and resumes?

I would argue its a very positive thing to include on your resume...

If your experience in the military can translate in any way to a job, you should definitely include it. At the very least, a positive discharge means you know how to follow orders...

Unless it directly has a connection to the job you're applying for, it's not necessary or wanted.

Many applications will ask your vet status. If you have experience in the military that might be relevant to the position you're applying for, mention it.

In general, mention it only if it might be relevant to the job.

It's hard to know in advance how some stranger, reviewing your resume, might react. Myself? I've got history as a civilian government contractor, and have worked with a lot of veterans. As a hiring manager reviewing resumes, I'd consider vet status as a "plus". But I can't speak for everybody.

Best of luck to you.