I recently bought a 6 can fridge cooled with a tec1-12705 unit, meaning the current rating is 5 amps. It gets about 7 degrees C inside the fridge and I want to upgrade it. I've been searching for a better cooling unit. Would a tec1-12708 for example, give better cooling results, if the voltage and amps are the same as my power supply?
There is a lot of difference in the various peltier devices. You could go here and compare what you have in your cooler and find a unit that supports a higher performance.
In all probability you could substitute a Tec1-12708 device, though be aware that the thickness of the various devices in the same size family will vary. Even 0.5mm can be important where you need to provide the correct pressure from the heatsinks on each side.
If you did replace a 12705 with a 12708 you would have to ensure that the wiring and any temperature regulator (if you have one on the cooler) or fuse would support an increase from around 5 A to greater than 8 A.