The non-instrument rated pilot reported that the cloud ceiling and visibility began to deteriorate about 10 minutes after his departure on a 60 nautical mile (nm) cross-country flight. The pilot located the nearest airport, a snow covered grass runway, and decided to land in lieu of continuing into the deteriorating weather conditions. He circled the airport to identify if the runway had any obstructions. There was a 65 foot wide plowed path in the runway that was bordered by snowdrifts. During landing rollout the airplane veered to the left after encountering a low lying snow bank that was obscured by drifting snow. The pilot attempted to regain directional control, but the nose landing gear separated from the airplane when it impacted a second snow bank. The airplane turned 180 degrees and came to rest in a nose down attitude. The nose landing gear and left wing were damaged during the accident. Prior to departure, the pilot obtained weather information for the planned route of flight using a commercial weather service. According to a Terminal Aerodrome Forecast (TAF), visual flight rules (VFR) conditions were expected for the duration of the accident flight. The nearest weather reporting station to the accident site, located about 6 nm to the south, reported ground visibilities greater than 10 statute miles (sm) and cloud ceilings 15,000 to 20,000 feet above ground level (agl), during the seven hour period before the departure time. However, about 10 minutes after the departure time, the ground visibility and cloud ceiling had dropped to 1-3/4 sm and 3,600 feet agl, respectively. The weather station reported light snow conditions in the area. The lower cloud ceilings and visibilities were not forecast to begin until after the accident flight was planned to reach the intended destination.