The recipes in for meatballs and hamburgers have their variations, but a lot of overlap. Many look almost the same. Other than the cooking method and size, is there a fundamental difference that sets them apart?
In the US, hamburgers are usually flat patties weighing somewhere between 3 and 8oz. and typically 100% ground beef. Many variations are possible, including mixing spices and other ingredients into the meat, but binders such as egg and breadcrumbs are not common. The defining characteristics of a proper hamburger for most Americans are the shape (flat), approximate size, beef as the primary ingredient, and sandwich-style presentation. A ground meat patty not made from beef is likely to be named differently (lamb burger, turkey burger, etc., even though the name 'hamburger' has nothing to do with ham).
Meatballs are smaller (maybe 0.5 to 3oz), roughly spherical, more likely to be seasoned, much more likely to contain eggs and/or breadcrumbs as binders, made from just about any kind of meat, cooked differently, served differently.
Also, the cultural traditions surrounding each are completely different. Spaghetti with meatballs and tomato sauce is classic comfort food; spaghetti with hamburger is unthinkable. Meatballs are usually served in a sauce of some kind; burgers (beef or otherwise) are served on a bun or bread with mayo, ketchup, mustard, melted cheese, lettuce, tomato, and so on. Many people like their hamburgers (beef) cooked rare or medium rare, but meatballs are always fully cooked. Meatballs can be held at temperature for a long time, or even cooked, chilled, and reheated; only cafeterias and other low-quality, high volume restaurants would pre-cook and reheat a hamburger -- at home or at a good restaurant the burger would be served as soon as possible after cooking.
Probably depends on the meatball. Yes, meatballs can be just round hamburgers, but for example, you can make meatballs for spaghetti that have eggs, onion powder, garlic powder, basil, oregano, salt and pepper and you will get a much better spaghetti from it than if you just make miniature spherical hamburgers.