As far as I can tell, admittedly based on my limited experience, the assignment of the roles seems random at best. But there are a few differences between trainers and breeders, more than may seem at first! While both classes can go through the badge program and fight gyms, only trainers need proof of badges to use HMs, powerful moves that Pokemon can learn that can also help them traverse the landscape easier; breeders can use HMs without needing badges.

Additionally, while there's the obvious to consider like you said, breeders have a higher chance of producing more remarkable Pokemon from those eggs than trainers do. The Pokemon caught by each class also grow and learn at different rates, depending on a few different factors.