The original mozzarella cheese tradition goes to the little town of Aversa, in Italy and it has a protected designation of origin called "Mozzarella di Bufala Campana". Then, having of course cows, they prefer to use buffalo for the elaboration.
Anyway it looks like that for creating mozzarella the key point is the procedure, not strictly the milk as there are varieties with cow milk in multiple countries.
You could enssure that the buffalo milk doesn't taste like the others since each milk provides different and fundamental nuances that affect not only taste but also the texture. Furthermore even different nutrition in the buffaloes, or genetic factors could create more differences between the same kind of milk as it usually happens with the widely extended cow milk, where each brand has a characteristic taste.
Sometimes, cooking is a matter of multiple little details here and there that create a better final result. Perhaps the original mozzarella helps in this aspect, so searching for "Mozzarella di Bufala Campana" in the package should ensure to have the original mozzarella in your house.
The elaboration with buffalo milk requires less milk for producing the same quantity of cheese compared with cow milk. In terms of nutrition, the buffalo milk provides more grease but less cholesterol, with a whiter color. It also compares better in terms of calories per unit of intake.
There's who affirms that the differences are almost unperceivable, but also the contrary although the price is higher with buffalo milk most of the time so the user has the final decision as always.