There is a standing joke among our Iron Blosam group to get the T-shirts and sunscreen out when I am there, and get ready for it to dump when I leave.
Since I do have strong impressions of it being warm on my March Utah trips, I decided to look it up. Updates after 2007 in italics. I have skied 3-4 days in LCC during Week 10 [date range March 5-18] for 12 of the past 13 years. Before that I had one trip at the same time in 1981 and 4 trips (1986, 1989, 1990 and 1994) in late March.
5 of the 11 Iron Blosam trips and 3 of the late March trips fit the pattern of this year: started out mostly packed powder, sometimes with a few scraps of fresh, but ended warm with no more than 30% of Snowbird still packed powder. In 1998 and 2004 the hot weather was enough to trigger wet slides that temporarily blocked the LCC road. Snowbird has by far the best snow preservation in Utah. It is safe to say that in all of these 8 years (half of the trips total) most of the snow at the Park City areas would have been slop by the end of my trip. What is strange, and thus cautionary in terms of drawing conclusions, is that none of my 11 Iron Blosam trips started out with the warm weather and spring conditions. The 1990 late March trip was springlike the first day; that's why it was the other time I climbed Baldy and skied Main Chute. It snowed a few inches the next day and went back to spring after that.
2 trips (1996 Iron Blosam and 1989 late March) trip started with considerable new snow and ended with Snowbird about half spring conditions.
3 Iron Blosam trips plus the 1981 trip had minimal new snow but were cool enough to stay mostly packed powder all 4 days.
2 Iron Blosam trips had about a foot of new snow sometime during the 4 days.
2006 at the Iron Blosam and 2002 (the year I missed, naturally) were powderfests with multiple dumps.
Conclusions:
1) Trust long term data over personal experience, even 17 years of it. LCC averages 90 inches of new snow in March, so the odds of powder are as good as any time. And the base is ALWAYS adequate, even in 1977. My Iron Blosam group has never seen the early season base situation which makes Alta ski much better than Snowbird. And they have seen a lot of the warm weather which makes Snowbird preferable to Alta.
2) If it doesn't snow Utah is usually warm enough in March that you will see a transition to spring conditions in just a few days. If you don't like that, come in January or February. I do not recommend staying in Park City later than mid-March because when it's good spring skiing in the Cottonwoods, it's much slushier at Park City.
3) If a packed powder surface is a higher priority for you than fresh snow, there are several Colorado resorts (Aspen, Copper, A-Basin, Crested Butte, Telluride, probably Taos in New Mexico also) that preserve snow better than anyone in Utah. Also, Adam was at Mammoth last weekend in similar warm weather, and he reports that the top of the mountain plus chairs 3 and 5 still retain dry winter snow. This is at least 1/3 of Mammoth's terrain, vs. the 10-15% of Snowbird that has not yet turned to spring conditions. And you need to be comfortable with steeps and trees to get at that 10-15% at Snowbird.