The headers should not be that much of an additional pain for him to the the transmission off. If you have stock manifolds and someone has put an H or X pipe on the exhaust, which is really common, the exhaust will have to come out anyways. With header you will most probably need to remove the exhaust to be able to at least loosen one side of the headers to be able to get the transmission out. It is a more work, so he will need to bill you more for it, that is why he is asking.
Difference is headers make more power, and weigh less, but when it comes to working on the car, like removing the transmission, they can be a hindrance, and will require additional work. Headers will not last a s long as manifolds, unless they are ceramic coated inside and outside, and usually hang way lower to the ground, so ground clearance, especially with a lowered car can be an issue. Headers emanate more heat and in some cars that can create issues with "cooking the starter".