They are almost interchangeable, however ''özür dilerim'' is used more frequently and also expresses somekind of guilt and seriousness. We would frequently use özür dilerim as a way to express that we admit our fault and ask for some kind of forgiveness.I am not sure but I think üzgünüm kind of lacks that admission of fault. There is also the expression ''kusura bakma'' which means sorry too.It is mostly used in a group of friends for courtesy. We would say something like ''kusura bakma seni yordum'' for courtesy.Which means sorry for the trouble. You can use it too as another way to say sorry but wouldn't sound intimate.