Here’s what I concluded:
An alias is often used for anonymity or legal reasons.
A pseudonym is specifically used by authors or artists to create a separate identity for their work.
A nickname is an informal, affectionate name used in casual contexts.
According to Wikipedia, pseudonym and alias are the same.
I would say pseudonym is often repeated across websites and can have some meaning. An alias is more random.
I would argue that something like a username represents a pseudonym given that it is not intended to replace an actual name, but used in place of ones name in communication like on this forum.
An alias on the other hand would be giving out false identities to people you meet who believe it to be your real identity. The same could apply to giving a website a fake name.
So a pseudonym isn’t intended to replace a name, only to avoid revealing it. Aliases on the other hand are fictitious names.