Mild is use here are it is referring to temperature. 'Mild winter' means a winter that is not too cold. You can also say 'mild tea' which means tea that is not too strong. Calm on the other hand is the opposite of rough. Have you been at sea when it is rough? The boat rocks a lot!

Basically, the word "calm" sounds more specific that "mild". Describing an entire season as "calm" implies that every day during that season was calm, whereas saying that the season was "mild" is a more general description. "Mild" can mean medium or average - by saying that the Winter will be mild, you're saying that it will be like most normal Winters, and will probably be as cold or as rainy as it usually is. The word "calm" is not really a measurement in the same way that "mild" is - it's more of a descriptive adjective that does not measure the severity of a season in the same way that "mild" does. You can use either word in that sentence, but mild makes more sense :)