Yes it can change the response, depends partly on how good the folk are that do the work, but that would be a quality issue more than a technical one.
Ultimately, yes it will make some sort of change, but you are unlikely to hear it and it would take some fairly sophisitcated equipment to measure the differant frequency response.
By this age, the type of speaker surround material is of less significance than replacing the capacitors in the crossover network, you will be surprised at the differance this will make.
Go for rubber surrounds.