The main take-away from this answer: notation is not always standardized, and it's important to make sure you understand in whatever context you're working.

The difference between a symbol and a sign is that a symbol can convey a deeper and more complex meaning than a sign. A sign is an indicator or marker for something very specific, very concrete and, in general, unambiguous in meaning.

Road markers are called signs because they usually convey something very specific - the speed limit, "Stop", a street name, etc. There is nothing open to interpretation with these markers (except in the minds of some lawyers, perhaps :-) ).

On the other hand a symbol, whether a cross, a dove, a ring, etc. can have a complex meaning and nuance that differs from one person to another depending on their experience, culture, upbringing, schooling, and so on. A symbol conveys a message of deeper meaning and is open to interpretation. A picture containing a dove could be meant to convey a message about peace, or it could be a picture of a bird. A ring worn on the finger could mean a commitment to another person in marriage, and marriage itself means different things to different people, but it may also be just a piece of jewellery.

So the "$" is called a dollar sign, not a dollar symbol, because has a quite distinct and concrete meaning. It represents a unit of currency, however, it doesn't mean any unit of currency but very specifically the unit of currency known as the dollar.

So, a "sign" is some unit of communication that represents something specific, while a "symbol" is a unit of communication imbued with deeper and more complex meaning.