Does the weight of a weapon affect Sneaking, or only its size category?
It's the category of the item that effects your sneak and not it's actual encumbrance weight. It's stated here that daggers are silent even though they do add to your total encumbrance - this would mean it can't be based on that.
Heavy armor and large weapons make more noise; daggers are silent, meaning that enemies are not alerted to your dagger swings, even missed ones.
Source: Eldar Scrolls Wiki