Latkes frequently include onions, eggs and matzah meal (or sometimes just flour). I frequently add carrots to my latke recipes. I have never heard of hash browns including carrots. Also, latkes are generally fried in vegetable oil. Hash browns may include onions, but the only other ingredient I have heard of is added potato starch as a binder, not eggs. In addition, hash browns are frequently fried in bacon grease, which is definitely not acceptable for latkes.