The difference between Cat5 and Cat5e cable can sometimes be zero. Cat5e cable is more or less identical to Cat5 (and in some cases, cable labeled as Cat5 can be identical to Cat5e). The difference on it is the standards. Cat5e has additional specificiations that must be met, and tighter control over losses and noise attenuation. At lot of Cat5 cable will meet the the specs, but not always guaranteed. Cat5e will.
Here's a good site explaining it: CAT5, CAT5e, CAT6, CAT7 and CAT7a Information
As for the bandwidth v synch speed - this is different for ethernet 100BaseT and 1000BaseT, compared simple binary signals, where bandwidth is essentially just the synchronising frequency. Fast and Gigabit ethernet uses multi-level signalling, so each wire can sent 5 different values (-2 Volts, -1 V, 0V, 1V, and 2V - representing something like 000, 001, 010, 011, and 100 in binary). Gigabit ethernet also uses additional twisted pairs compared to fast ethernet.
Now, this is about as much as I know for sure about the two standards. But, a little bit of thinking about these multi-voltage levels, and having addition pairs for use, means, say for 2 pairs where only 1 wire of each pair is used for signalling, 25 (5 x 5 voltage levels) different discrete voltage signals are available. So, that 100Mhz becomes a theoretical bandwidth of 2.5Ghz in a traditional sense.
I am by no means an expert on ethernet, but I know a bit about digital electronics, and I'm comfortable in saying that the above is essentially what is going on with ethernet - or something very similar. I don't know how they impliment the coding with the different voltage levels over more than one pair, it may not be 5x5, but the fact is they do use multi-level signalling and use more pairs on 1Gbps, so it will be using a system similar to the above.
Hope that helps.