Learn English – Difference between ‘Worth less’ and ‘Worthless?’

Cooperating means working with someone in the sense of enabling: making them more able to do something (typically by providing information or resources they wouldn't otherwise have).

Collaborating means actually working alongside someone (from Latin laborare: to work) to achieve something.

The confusion comes from the overloaded meaning of "work with": In the "Work with me, people" sense, it means to go along with my idea - it's a passive condoning or suspension of disbelief rather than an active involvement. In the "I'm stuck, can you work with me on this problem?" sense it is a request for active commitment.

So in terms of helping achieve something, the ordering is something like collaboration, then cooperation, then passive indifference, then active obstruction.

Surely there should be more of a difference between the two words. If they mean the same thing surely there would be no need anymore for two words

There are a lot of near synonyms in English, so this does not follow at all. It's particularly so when you consider that act entered the English language in the late Middle English period, from French, while deed was in the language from the very beginnings of Old English. Many, but not all, seeming redundancies in English vocabulary have a similar origin story.

It is true that there are differences in nuances. The primary one would be that an act would generally be need to be relatively remarkable before we would use the term deed. One can do a particularly brave or noble or dastardly or evil deed, but one can't really do a lazy or everyday or common deed. There's nothing technically wrong with the latter, but it wouldn't be an idiomatic use.

There are similarly some differences in what verbs and prepositions are used with either. One doesn't do an act, one cannot be in the deed of anything.