They're both in the same class of drugs, and pretty much treat the same symptoms. Paxil is a newer medication, and if the dosage isn't working you may need to have your doctor increase it. Prozac is an old drug and I think the newer ones come with fewer side effects. It's best to discuss this with your doctor since he is aware of what will best suit your needs.
Recent study is showing that Paxil causes a change in personality in term of reduced neurotic symptoms and increased extraversion.
Paxil appears to have a specific pharmacological effect on personality that is distinct from it's effect on depression. If replicated, this pattern would disconfirm the state effect hypothesis and instead support the notion that SSRI's effects on personality go beyond and perhaps contribute to their antidepressant effect
For more information on the study: http://www.ncbi.nlm.nih.gov/pubmed/19996037
Prozac stays in the body a long time, whereas Paxil leaves the body very quickly. That means Prozac is much less likely to have withdrawal effects than Paxil, but then, so do most meds! If you're treating depression, they are theoretically interchangeable, but the reality is, everyone is different. Paxil seems to be stronger than other ssris, possibly because of its larger effect on choline receptors, which accounts for some its difficulties. Prozac, of the ssris, has the least weight gain and is easiest to quit. Paxil seems to work better for anxiety, but there are other ones you can try as well if you don't want to do Paxil. And nobody can tell you which will work for you -- it's just too individual.