The non-instrument rated private pilot diverted to the alternate airport so as to avoid flight into instrument meteorological conditions along the intended route of flight path to a destination airport. While landing at the alternate airport, the airplane bounced and departed off the side of the runway. The airplane sustained substantial damage that included the engine firewall and right wing. The pilot stated that he reduced engine power and airspeed so as to increase separation with a preceding aircraft that was also landing, which resulted in "too much" airspeed being reduced and the bounced landing. The pilot stated that he was using active noise reduction headsets for the first time, which "greatly" reduced engine noise and "muted" audio input so as to not receive the "normal" audio indications that his senses were accustomed to on prior flights. The accident was not reported by the pilot and was discovered by a Federal Aviation Administration inspector during routine surveillance at the airport.