As far as I see the main difference is that Assessments are there to discover existing facts (character aspects as well as those already established by another player [mostly by the GM, probably]), Declarations are there to create new ones.
If the aspect you want to declare is unusual or unfitting for the scene you're in, the roll to declare it might be quite high. Why would there be a bucket of water in a lawyers office? Well, roll a 10 and tell us (or don't even roll, if the group decides it doesn't make sense).
And even if you can declare something, it might not be automatically accessible. For example, instead of a bucket of water in the office, you declare that way down the hall from the lawyers office is a janitors closet, and there is a bucket with water that the cleaning crew didn't empty. You as a player have declared that, but your character would still have to find that bucket, so you do an alertness roll and describe how your character hectically runs down the hall to find a bathroom or something and ends up in the janitors closet.
And declarations might be resisted as well. If you want to declare that the nameless mercenary has a "trick knee", he should be able to resist that. Since nobody really thought about the mercenary's knee up to this point, he doesn't have an aspect for that. It is a new fact, a declaration, not an assessment about a character aspect. But since you are targeting a character, he would get to roll against it.
In the end, I think declarations are mostly there to remove the typical "Is there a...? Or maybe a...? Or does he have...?" questions from players. Instead, they can say "There is a ... that I can use to...!" and just go ahead and use it, if the . Assessments ARE the typical "Is there a...?" question.
Though I have one question myself: What skill do you roll for most of the declarations like "bucket of water" and the likes? I usually roll alertness as in the "Let's see if you see it", but that seems rather weird. It kind of feels like it might be better to treat this as a "create a fact" thing and spend a fate point.