First off, let the record state that I am biased as I am in Puerto Viejo and run a business...
Now that we have the formalities out of the way...
It is a bit of apples to oranges as they are definitely different places.
Puerto Viejo is less developed and would would be ideal for more adventurous type travelers who are looking to really experience tropical jungle/beach type of environment.
Accommodation on Caribbean side tends to be a bit more rustic and less North Americanized, For some that is great... But for others, that is a total drawback.
As for cultures, the Caribbean has quite a bit more diversity in types of people that live there then Manuel Antonio. Before it started being a haven for foreigners, there was already 3 distinct cultures living there before their arrival. Indigenous, Hispanics and Afro-Caribbean are evident all around Puerto Viejo.
As far as beaches go... Caribbean is basically one long beach which stretches from Panama to Nicaragua. The beaches south of Puerto Viejo tend to be most popular with their whiter sand colour.
Weather for Caribbean coast is very different then Pacific. Rather then explain the differences here, you can get a detailed overview of what one might expect here. http://puertoviejosatellite.com/weather.php
If you looking for a total "sun vacation", Puerto Viejo or Caribbean side would not be for you. Then again, you might rethink Costa Rica as your choice as rain is responsible for all its lush environment that many people came to experience.
Cost wise, Yes, Caribbean will be cheaper, but not a huge amount. It is easy to do all desired activities in Caribbean side without a car which would save a huge amount of $.
If you want to see Caribbean through other travellers eyes, rather then polished hotel photos, I would suggest you check out these photo galleries. puertoviejosatellite.com/photos-blogs.php…
Well that wraps up what I have to say. Either place, I am sure you will have a great time. Your kids are right age that if they are adventurous type will enjoy this type of Vacation.