No, there is no biological difference between the Pfizer or Moderna. Both vaccines can cause various adverse effects, but these reactions are reported to be less frequent in the Pfizer/BioNTech vaccine compared to the Moderna COVID-19 vaccine; however, the Moderna vaccine compared to the Pfizer vaccine is easier to transport and store because it is less temperature sensitive.
If you get side effects from the vaccine it's generally very mild, maybe a slight fever for a few hours, a sore arm (which most IM injections cause) or some diarrhea. That's it, and that's actually a great sign that your body is creating antibodies to the spike protein found in Covid-19 on this spike protein is harmless, the SARS-CoV-2. spike protein is deadly.
If you're considering which to take, the Moderna is more often offered at places like walk in health clinics or pharmacies because it's easier to store. Because of the low temperature required to store the Pfizer, it's more likely going to be given at a hospital or any large facility that has the freezing capability required for the Pfizer vaccine.
Incidentally as far as "testing" of these vaccines, they have all been tested for at least six months prior to public release on many thousands of people. My nephew was in the Pfizer study and he has Guillain-Barre syndrome. The FDA had tons of data and statistics to use before submitting it to the FDA for emergency use approval. Since the vaccines have been approved by the FDA for emergency use, over 4.6 billion doses have been put into arms. I'd say that's a pretty good example of thorough testing if you want to call it that.