Advil and Motrin are brands of ibuprofen. The only consideration when choosing a brand for ibuprofen is the safety record of the manufacturer to gauge how trustworthy they are.
The Advil brand is owned by Pfizer and the Motrin brand is owned by Johnson & Johnson. Other ibuprofen brands include Nurofen, Caldolor, EmuProfen and Brufen.
Ibuprofen is the generic name for the active ingredient in all these drugs. So there is no difference between Motrin and Advil in how fast the drug acts, how long the effects last, the efficacy of the drug, or the side effects.