"At large" can be used more generally, for anything dangerous that is still free. For example : Two lions escaped from XXX zoo yesterday. One was recaptured immediately, but the other is still at large.
It can also be used humourously as in the children's book "Paddington at Large" which relates all the problems Paddington Bear manages to cause and how they are eventually resolved.
"On the run", however, definitely refers to a criminal hiding from, and being hunted by, the police. It couldn't, for example, be used in the sentence above describing the escaped lion - which is not consciously "hiding" - or in the title of the Paddington Bear book - although Paddington gets himself into a lot of trouble and is humorously "dangerous" in that respect, he's not "criminal" and the police aren't looking for him.
Susan53 had a great answer to your question. At large has more of a general connotation whereas on the run usually exclusively refers to criminal activity. So it wouldn't be entirely correct to say that zoo animals were on the run, but you could say a criminal is still at large.