I buy fresh cream and shake it off to get butter and buttermilk for baking. Would there be any difference if I add the fresh cream directly into the dough instead of the separated butter and buttermilk?
Yes, for two reasons. First, baking involves much more complex chemistry than other cooking, and can be sensitive to things like the the precise way in which things are mixed. Creaming butter and sugar, for example, is not just about combining the ingredients but about developing a texture. Things like development of protein strands and air bubbles and emulsions are vital.
Secondly, the "buttermilk" that remains from making butter isn't likely to be much like the buttermilk in your recipe. When a baking recipe calls for buttermilk, they probably mean cultured buttermilk, which is the leftover stuff with cultures added to it and allowed to ferment like yogurt.