Let's begin with some helpful definitions:

"Contaminated runway" means a runway of which a significant portion of the runway surface area (whether in isolated areas or not) within the length and width being used is covered by one or more of the substances listed under the runway surface condition descriptors. More than 25% of the runway surface area within the required length and width being used is covered by the following:

• surface water more than 3 mm (0.125 in) deep, or by slush or loose snow, equivalent to more than 3 mm (0.125 in) of water;
• snow that has been compressed into a solid mass that resists further compression and will hold together or break into lumps if picked up (compacted snow) or
• ice, including wet ice

"Wet Runway" - when the runway surface is covered with water, or equivalent, less than specified for a Contaminated Runway above, or when there is sufficient moisture on the runway surface to cause it to appear reflective but without significant areas of standing water.

"Damp Runway" - when the surface is not dry but when the moisture on it does not give it a shiny appearance.

"Dry Runway" - neither wet nor contaminated.

As far as our question is concerned, wet snow creates a slippery and slushy surface, making it more difficult for an aircraft to gain speed during takeoff. This increased rolling resistance means that more thrust is required to achieve the necessary speed for lift-off or available rolling distance for take off.