Gigabit Ethernet is really pushing the limit of Cat5e cabling. Cat6 is superior to Cat5e when it comes to insertion loss, near end crosstalk (NEXT), return loss, and equal level far end crosstalk (ELFEXT).
Consequently, Gigabit Ethernet over Cat6 could be marginally faster than Cat5e, by the amount of errors produced over Cat5e, even though the networking equipment on either end still operates at the same 1 Gbit. This difference will vary based on other factors in your environment, and also networking equipment. In real world terms, the difference is probably too small too measure without sophisticated tools, and so small as to be made non-existent when other delays introduced by operating systems, switches, routers, and end-user applications are considered.
Most companies are installing Cat6 now for new installations (some reports say as much as 90%). That is mainly for future proofing, since the labor to run cabling can often exceed the cost of cabling itself. I would recommend Cat6 for all new installations, but not to replace existing Cat5e.