the main difference is what is acting as the selection pressure.
For natural selection, take the example of grey and black moths in an environment, the selection pressure is the predators who may see the grey moths better when they are against darker bark on trees. This will allow for the moths to be 'fitter' since they are selected by their environment.
for artificial selection think of cows being bred to produce more meat. Humans are the selection pressure selecting more meaty cows to breed together. In the end, they aren't necessarily 'fitter'.