The modules will have some effect on your performance; but this is only as far as range, rejection to strong interferers, and the like. You have to step up in price quite a bit to get modules which really make a great deal of difference (i.e. pre-selectors, LNAs, XTAL filters, or zero-IF sampling). But modulation characteristics does play some role on performance.
  • AM/ASK modules are always at the lower end of the perfomance spectrum due to their susceptibility to noise.
  • FM/FSK modules are less susceptible noise
  • PM/PSK modules are usually even better
For most applications which I have seen here the bulk of the improvement comes from the protocol used. The better the protocol the better the performance (i.e. reliability and quality of the link). Some modules have built-in protocols, some not.