If you look at a map of West Yellowstone NP, Fishing Bridge is in the far eastern part and the town of West Yellowstone ie in the far west, it is a wash. Staying at Fishing Bridge is in the woods in close sites and you will pay the high prices for supplies that is controlled thru out the park. Staying at Grizzly RV Park gives you well manicured park, that is well patrolled, no nonsence, clean, secure, hot showers, multiple clean laundry rooms. The town of West Yellowstone gives you a break from the park, competive pricing on gas/diesel/propane, ice cream parlores, grocery stores, yes more than 1, IMAX Theater, sporting good shops, pizza, Mexican, etc. town of West Yellowstone gives you choices.
Enjoy your stay there.