It depends on the car. On some cars it makes a helluva difference. For instance my 61 Skylark, I removed the drip rails and it really improved the look. 99.9% of the people don't realize I shaved them (but then again 85% of the people don't even know what the car is).
As far as rain and wind noise it will also depend on the cars design. With my car the roof and A pillar over hang the door frame, so no wind noise and no water would build up (if I drove it in the rain). Some cars/truck get horrible wind noise if the drip rails are removed. If your door/window/window frame are even with the roof or A pillar there is a good chance you will have issues with wind noise.
To the OP, it is a very tedious job and lots of potential to go wrong. Once you start you are commited, no turning back. It is not something you want to try to do in an afternoon. Take your time and only weld 3/4 - 1" at a time and skip around. Cut a little, weld, move to another area, repeat. I used a portaband and a body saw. I TIG welded mine and it could not have been done with a MIG because the area was so thin with the way my car was designed.