Barrington is close to the bike path, you'll be on the east side of the bay. Seekonk will be the shopping area. Roads in and out of the East Bay side tend to be slow speed limits and only one lane each way (if you're headed to Bristol or Newport). Beaches would be limited - unless you drive to Newport. You would be closer to Cape Cod. Not sure about the restaurants there.
East Greenwich is on the West Bay - it's close to Route 95, which makes it accessible to Providence, CT, Warwick (known for its shopping on Rte 2). Roads tend to be 2 lanes each way and have higher speed limits. Beaches are plentiful, depending on where you want to go - Narragansett is 20 minutes away and has the best beaches in RI. East Greenwich has excellent restaurants. I favor the West Bay - easy access to everything.