That's a good question. For me it's not really about it running all the time but it's about not having to spend any mental effort to switching "mode"and running the tests. With this and the IDE integration it really makes testing a first class activity. I think wallaby that I linked has some videos of the experience which might make more sense than my words.
Most continuous test runners don't actually run on every key stroke but wait until it's syntactically valid. It's also common for them to do some smart diff and only re-run a subset of the tests.
I absolutely call them vegetables. It's a kitchen term and it absolutely makes sense to categorise them alongside tomatoes, beans, carrots, squash and cabbage. People get too hung up on things only belonging to exactly one category.