Proofs are impossible in many cases because nobody really fully understands the requirements, they are sort of working them out as they go. (and in any case they will change over time). That might be what you meant by devilishly hard to write.
Tests let you instead say "with this setup here is what happens", and then ensure that whatever else you change you don't break that one thing.
To my knowledge nobody has scaled proofs to very large problems. I still think proofs should have a place in better code, but I can't figure out how to prove anything in my real world code base. (I could probably prove my languages basic containers - something that itself would be valuable!)
