The primary difference between a dumpling and a noodle, besides shape, is leavening.
Dumplings usually have either egg or baking powder to make them lighter.
There's considerable bleedover in terminology here.
It's reminiscent of the difficulty of defining "chowder".
You're always finding a counterexample.
I wouldn't be surprised if someone came up with something called gnocchi, gnochs, gnocchetti, that has no leavening.
I can think of two myself.
Gnocchetti alla romana are semolina "dumplings" sometimes leavened with egg, particularly when they contain spinach, but more often not.
Gnocchetti sardi are dry pasta shapes that kind of look like thin potato gnocchi.
cooking.stackexchange.com/users/1664/Rich Armstrong
Noodles are thin and elongated.
Dumplings are not.
Both are unleavened dough cooked (frequently) in boiling water, but generally the distinction tends to fall along shape-based lines.
cooking.stackexchange.com/users/5130/tkone
Possibly because they aren't normally dried out.
It's probably just 'one of those things'
- I don't think there is an ISO committee of food naming
cooking.stackexchange.com/users/4483/Martin Beckett
Perhaps because gnocchi is cooked in the same way to a dumpling, by boiling in something like hot water/stock or steamed.
cooking.stackexchange.com/users/5162/nixy 
There are in general two main classes of dumplings.
Potato based and leavened dough based.
Gnocchi generally fall under the first category along with traditional central-european potato dumplings, some sorts of German "spatzle", Slovak "halusky" etc.
Second sort of dumplings is based ond dough, sometimes strictly flour/egg based, sometimes enriched with buns, but almost always leavened.
Other generally sorting rule that apply is the dough thickness.
All the pasta/noodles are made of THIN plates of dough, whether dumplings are boiled either as balls, droppings of fresh dough directly into boiling water, or long cylinders, that are sliced upon serving.
cooking.stackexchange.com/users/None/
Gnocchi can have potatoes in them; I don't think noodles ever do.
cooking.stackexchange.com/users/4861/tim
I've recently been told that gnocchi is technically a dumpling, not a noodle. What about it causes gnocchi to fall under the dumpling category?
Why is gnocchi a dumpling and not a noodle?
cooking.stackexchange.com
<noodles><dumplings><gnocchi>
cooking.stackexchange.com/questions/12937
cooking.stackexchange.com/users/3096/fbrereto
Noodles have a different shape and do not contain potatoes, but both are unleavened dough cooked in boiling water. Noodles are long and thin. Gnocchi has potatoes in.
Noodles are thin and long. Both contain unleavened dough and are cooked in boiling water. Gnocchi has potatoes in which is the major distance.