Juan les Pins vs Antibes with small children?

There are exceptions to the generalities everywhere -- Antibes has its share of touristy spots, and there are plenty of old-timers rolling boules in Juan les Pins -- but overall, Juan les Pins is definitely a modern adult resort area and Antibes is historic and catering to the elderly/tasteful.
One small thing to keep in mind is that the beach in J-l-P is sandy and green, good for actual bathing and playing but not as picturesque, while the beach in Antibes is more Nice-ey, stony and blue -- great for photos and sitting there, not so much for strolling/playing.
I had a conversation with the co-owner of one of the RS recommended restaurants in Antibes, the one that is located in the cooperative in a little basin, who said that he had worked for years for one of the big restaurant impresarios in J-l-P, but left when it became too clear that the impresario was putting profits before people. He and the other staff, I realized, were having fun by parodying the way staff in J-l-P drum up business with a little bit of sing-song and sales patter.
In J-l-P the wait staff will speak to you in whatever language you speak; at this place in Antibes, when the co-owner realized I could get by in French, he slapped me and said ~if you know how to say it in French, then say it in French!~

So, I lean towards Antibes overall, but one last thing I will mention is that I never found a really authentic bakery (i.e. mostly local clientele) but I did find one in J-l-P. I probably just needed to do some more looking.
There were plenty of families in the restaurants of J-l-P, with strollers and carriers of various kinds not quite tucked away from the aisles. This was in July/August, when there are kids everywhere.

I preferred Juan-les-Pins to Antibes. It definitely does not feel like an "exclusive" place, if that's what's concerning you. And as said above, it has a sandy beach rather than a rocky one.