Why do computers only use 0 and 1?
It wouldn't speed them up. Now it's easy: to make a basic logic gate like a NAND the logic inputs either pull the output to Vdd or to ground. If you would use intermediate levels you would need FETs to go to levels like Vdd/2 or Vdd/4. This would consume more power, and would require more accurately working components, which would need more time to settle to the final level. If you would stuff more values in a single data unit the required accuracy would increase, as would settling time. The binary system used now just pushes the FET hard to Vcc.  

exscape mentions noise immunity, and that's what the accuracy refers to: how much may the signal deviate from nominal. In a binary system that may be almost 50 %, or more than 0.5 V in a 1.2 V processor. If you use 4 different levels they're only 300 mV apart, then noise immunity can't be better than 150 mV, possible 100 mV.

Note that there are Flash devices which use multiple levels to store more than 1 bit in a single memory cell, that's MLC (Multi-Level Cell) Flash. That doesn't increase speed, but packs more data on a single chip.