Other than both being sparkling wine, no.
Champagne is undoubtedly the king of sparkling wine. Three things must be true for a sparkling wine to be called Champagne. First, it must be produced in the Champagne region of France…no ifs, ands, or buts. It must also be made from Chardonnay, Pinot Noir, or Pinot Meunier grapes. Finally, Champagne can only be fermented using the Champenoise method which is a two step fermentation process: first in barrels, then in bottles. Anything else, even in France, is just sparkling wine.
None of these things are true about Prosecco. First of all, Prosecco is named for both the grape (now known as Glera) and area in the Veneto region of Italy where it is produced. To be a true Prosecco, at least 85% of this grape must be used. Fermentation is done using the Charmat method which is also a two step fermentation, but both take place in stainless steel vats.
Prosecco is meant to be drunk when it is young and fresh, so the production process is usually faster than that of Champagne. That is why most Prosecco is modestly priced while finer Champagnes can be quite expensive.