What is the difference between these two sentences? "If A is true, then B is true" and "Since A is true, B is true"

To answer the second question first, then is optional after if; also after since, but not as common.

The first question, however, involves an important distinction. Whereas if is a standard hypothetical, since has a Presupposition.

A presupposition is something that has to be true in order for the sentence to make sense. If I tell you that

• Bill's brother is visiting me

then you have to believe

• Bill has a brother

in order to understand what I told you.

One feature of presuppositions is that they can't be negated. If I tell you

• Bill's brother isn't visiting me

you still have to believe Bill has a brother to understand it.

This property of presuppositions distinguishes them from assertions like Bill has a brother, which can be denied:

• Bill has a brother ~ Bill doesn't have a brother.

Language is full of presuppositions -- they're one kind of "connotation", which is a cover term for a variety of phenomena like this. Presuppositions distinguish and from but, and used to be from isn't any more, and accuse from criticize, and many more.

Since means 'if it's true, and we all know it's true, that ...'; it's used to consider implications, while restricting discussion of details of the presupposed clause. Essentially it says the material in the clause it introduces is background; nothing to see here, move along.

That's why presuppositions are prominent in all propaganda. If you can make someone think that everyone else believes X, they are likely to conclude that they must believe X, too.