They both look similar, and aren't they both washbowls? What's the difference between them, or are they just synonyms?
My understanding is that a 洗面器 only refers to small, shallow and plastic (or sometimes clay) ones, large enough to wash your face. You can buy this everywhere.
たらい can be made of anything, and it's small or large (enough to contain a person). Perhaps the only requirement is that it has to be relatively shallow. I think it safely includes 洗面器, and I can imagine someone who never uses 洗面器 in favor of たらい. Especially metallic or wooden ones are almost always called たらい.
Deep one with a handle is called 桶 or バケツ.
洗面器 is a basin that should be about 1 foot diameter according to the necessary volume of water.
たらい is a tub, generally Japanese associates 洗濯用たらい which is about 3 feet diameter to be capable to wash clothes.
A small wash bowl.
Usually portable and can be brought to public bath places. Mostly used to pour water on yourself. Can also in some cases refer to a wash basin to wash your hands and face.
This is a bit peripheral, but I think interesting. The well-known economist Morishima Michio notes in his memoirs that when he served in the Imperial Japanese Navy a タライ was called オスタプ, which was the English word "washtub", picked up in the late nineteenth century, when the British Navy was involved in training the Japanese Navy. Whether the word has survived into the Maritime Self-defence Force I don't know.