Pretty much everything is different between them. Pleasure saddles have a very small (and mostly decorative...i/e useless) horn, they usually have low cantles and very small swells. Trying to run barrels in one of those would be not so different than running barrels bareback.
Barrel saddles have tall horns that are easy to grab in a hurry, they have high cantles and larger swells for the security while going around hard and fast turns. However, that type of saddle would look out of place in a pleasure class.