Since both are in the Basque region, you will find them to be almost identical in terms of architecture, food, and culture to much of the Spanish Basque region----the only thing really "french" about the French Basque region is that the people who aren't speaking Basque speak french. So if you want a taste of France you might consider staying somewhere outside of the Basque region.
That said, if these are your choices, they are both good ones. Biaritz might seem a bit more "french" because it is a tourist/beach destination, meaning it is a bit less in-your-face Basque. But that tourist destination also brings tourist culture, lots of Brits, etc etc, more than a quaint village or town. But it does have a nice beach.
SJPdP should prove to be a more quaint, rural, authentic, medieval basque town. But it is smaller, less to do, and full of pilgrims (who you will be around once you begin).
A third option might be Saint Jean de Luz. It is near Biaritz and offers many of its same features, but a bit less touristed, a bit more authentic, and a really neat basque city center.
As others have mentioned, Bayonne is also a good option. It might seem a tad more french because of its size, but I've always found it to still be very, very Basque.
Its just kind of a tough question to answer because the French Basque region has more in common with the Spanish Basque region than it does the rest of France (other than language).
Hope this helps!
