It indeed makes the request smoother as using the conditional makes it more, well,… conditional. It looks more like a wish, and less as an order, although depending on the people asking the constraint might be the same.

If I’m not mistaken, the same applies in English.

Yes, it is conditional: an expression of potentiality and a degree of uncertainty. I think when you state the two options, you miss the force of the first phrasing by writing it in its elided form. If you state it in its long form it becomes more obvious:

I would prefer that you did not address me as consul

Similarly it is politer to say:

Je voudrais un café

as opposed to:

Je veux un café

I have heard people use the latter in shops/stores, so presumably it is not appallingly rude, but the first form is definitely more polite.

Similary, if you wished to make a polite suggestion to someone that they should do something you would use the conditional of «devoir», e.g.

Vous devriez fermer votre fenêtre car il va neiger.