Cat5e has a max transfer rate of 350Mbps while Cat6 has max of 1Gbps.
Depending on the specs of the "old computer" I might not care about the difference, since a gigabit network card is constrained by the speed of the PCI bus as well.
Source: FAQs: CAT-5, CAT-5e, CAT-6, CAT-7 Patch Cables...