What is the difference between "The police are catching the thief" and "The police are chasing the thief"?
Firstly it should be 'are'
Chasing is the act of going after them, catching is what they hope to do at the end.
They may chase the thief who escapes, so they don't catch them.
Chasing is the ATTEMPT. Catching is the RESULT.
One (hopefully) leads to the other.
Chase: to pursue
Catch: to seize, capture
You can chase your ideals, you can't go catching them... So I'd say chasing doesn't have to be oriented towards earthly and tangible things, while catching has to be (since you can't really catch an idea)
As well as what mgb and Brendan have said: "catching" may involve "chasing", but does not necessarily do so. To "catch mice" usually means catching them in a trap.