What's the difference between the Merchant Crown's Finder and Keeper abilities? Also, how can I increase the success rate of these abilities?
According to Gamefaqs the keeper ability is just a more effective version of the finder ability.
While there doesn't seem to be a way to increase the success rate, there is a trick for point farming which can be combined with finder/keeper to farm gems too. By using two wayfarers and auto-battle you can immediately escape from any battle you enter and earn 5 points. If you have a merchant wearing Hermes Sandals set to use finder/keeper it will harvest a gem before you escape from battle. Go to somewhere with enemies, and tape down the d-pad so you're running into the wall. Come back later for free stuff. Source (Ctrl+F Point Harvesting Trick)
Both Finder and Keeper will attempt to find Gems during the battle. Keeper has a higher success rate than Finder, but Keeper costs 2AP whereas Finder costs 1AP.
In each battle, both Finder and Keeper can be successful once (with enough attempts). Each Merchant in the party can also be successful at both Finder and Keeper. Therefore, you can get 8 gems per battle with 4 Merchants.
If one party member is a Merchant, then there is a higher chance of gems dropping at the end of battle. More than one Merchant does not increase this chance.