If you feel resistence but the caliper is not moving, I almost bet the problem is inside the cable housing, most probably dirty/rusty, bent or frayed cable. I would recommend the following diagnostic procedure:
Detach the front end of the brake cable from the lever (most levers allow you to do so). Grab the cable "head" with one hand, and the housing with the other, and try to move the cable inside and outside the housing. You should feel the cable running almost free inside the housing, and the cable exiting from the other end.
It is important to distinguish between a single long cable housing from the lever to the brake, and many short stretches of housing, so as to test each stretch independently. If that is the case, you can free the housing from the frame braze-ons (allowed on most frames nowadays), and verify that the cable runs free inside all of them. If not, you found your problem.
Don't forget to check for the curved metal part if it happens to be a V-Brake.
Hope it helps.