Champagne is sparkling wine that is produced in the Champagne region of France. The grapes used in champagne are usually Pinot noir, Chardonnay and Pinot Meunier. I think that there are a few sparkling wines that are grown outside of Champagne, France, that are allowed to be labeled as champagne but for a purist, they aren't actually champagne. The politics are complicated, and you can get an overview of the idea at the champagne wikipedia page.
Spumante, on the other hand, is a sparkling white wine from the Piedmont region of Italy. (It is also called asti or asti spumante.) It's made from the Moscato Bianco grape. More information can be found on the spumante wikipedia page.
I find spumante wines to be very sweet, whereas you can purchase champagnes that are brut (dry), demi-sec (half-dry) and doux (sweet). Most champagnes are pretty dry. Champagnes are also typically sold at a premium, whereas you can find bottles of spumante for about $5.
I think it's a matter of taste for when you want to drink champagne vs. spumante. If the cost is an issue, then champagne should be saved for special occasions. Spumante is good for a meal or occasion that calls for a sweeter sparkling wine. If I want a dry sparkling wine and don't want to shell out for champagne, there is always Prosecco, Cava, or other generic sparkling wines.