What could cause severe hand pain and vein swelling?

Lots of things: Most important, if the pulses are not as strong as normal on both sides of the involved wrist, or the hand is very blue, seek immediate care (vascular emergency). Someone else can give you a full list of possible causes.

May be a blood clot: A blood clot in the veins draining blood from that arm should be considered. Please talk to your doctor right away.