When running gels what is the difference between constant volts or constant amps?
I'll answer only for SDS-Page, which is the system I am most familiar with.

With a discontinuous buffer system, such as the well-known Laemmli system, resistance increases during electrophoresis, as (very mobile) chloride ions are replaced by glycinate (glycine ions). 

From Ohms law:


  Voltage (V) = Current (I)  x Resistance (R)


and the definition of power (Watts): 


  Watts (W) = Current (I) x Voltage (V) = I2 x R



At constant current electrophoresis will proceed at a uniform rate, however, as the resistance increases, so too will the voltage and the amount of heat generated.  

In some discontinuous buffer systems, such as the Tricine/SDS system due to Schägger & von Jagow, this may be enough to crack the gel plates!

At constant voltage, the current will decrease during electrophoresis; as a result less heat will be generated than constant current electrophoresis, but the rate of migration of the sample will decrease with the decrease in current. 


References

Laemmli, U.K. (1970) Cleavage of structural proteins during the assembly of the head of bacteriophage T4. Nature 227, 680-685. [PubMed]

Schägger, H. &amp; von Jagow, G (1987).
Tricine-sodium dodecyl sulfate-polyacrylamide gel electrophoresis for the separation of proteins in the range from 1 to 100 kDa. Anal Biochem 166, 368-379.[PubMed]