The differences between each will vary by manufacturer. IN some cases, it's the number of twists AND the pair-pair twist within the sheath AND (in some cases) the guage of some of the conductors AND AND AND (so forth).
There is no one physical characteristic that puts the cable into a specific Category specification, it's generally a number of things.
The specs for each Category that provide increasing frequency tolerance are things like Near End Crosstalk (NeXT), Far End Crosstalk (FeXT), Attenuation, and the Attenuation-to-Crosstalk Ratio (ACR). There are other specs, like Power Sum, impedence +/-, etc ... but the above are "the biggies."
As long as the manufacturer can consistantly hit the numbers, their cable continues to be rated.
At Anixter, each new shipment of cable is randomly sampled and tested tested to see if it meets their "Levels" rating (the origin of the "Category" rating system). If the sample fails the "Levels" spec (which is tighter than "Category" spec) then the whole batch goes back to the manufacturer. Other distributers may do the same thing, but nothing that I've ever heard of.
The first few generations of GigE devices worked OK on Cat5, because the equipment manufacturers knew that that was most likely what would be available for installation. As Cat5e was ratified and became more available, that became the target spec.
Cat6 isn't "needed" for anything in data networking at this time, but the additional overhead provided makes up for some of the usual slop and unavoidable sub-optimal conditions in most / many environments.
Remember, structured cabling isn't just a data thing, the cable plant is good for much more than just Ethernet (i.e., video, broadband, serial, audio, generic low-voltage signaling (alarms, for example), even telephone (!!)).
FWIW
Scott