To me quadricycle is a more generic term that includes four-wheeled cargo-cycles, pedal cars and four-wheeled velomobiles.
If we're talking motor vehicles, then we're into legal definitions, in which case the difference between a quadricycle and a car is mostly weight and power output, and what category of licence you need to drive one. Quadricycles would include the 'quad bikes' beloved of farmers and oiks, and IIRC the Renault Twizy.