What exactly are the differences between "diligent", "assiduous" and "sedulous"?

There is no difference in core meaning, and even the connotational penumbrae overlap a lot. It's mostly about the attitude you're trying to convey; and this is going to depend more on each reader's actual experience with the words than with any broad consensus.

Here's how I would use them:

• Diligent if I wanted to express respect and admiration for the worker's application to her project
• Assiduous if I wanted to express respect for the worker's application and thoroughness, while leaving room to doubt the value of her work
• Sedulous if I wanted to acknowledge the worker's application and thoroughness, while leaving room to wonder whether the work might not be better served by insight and imagination

But that's a very personal view; others may feel differently.

Of the three synonyms, diligent is by far the most commonly used, in my experience. And sedulous is the least common — you'll get a lot of strange looks if you use it.

I use and hear diligent much more often than assiduous.

Assiduous is more about attentiveness/attention/attendance, and diligence is more about effort.

• assiduous
• diligent

Assiduous allows for improvement to rules rather than the 'blind' following that is diligence. It is more than diligence.