All four of these words can be used to refer to the reason for something, [...] They are not used in the same way.
1. As and since As and since are used when the reason is already known to the listener/reader, or when it is not the most important pasrt of the sentence. As- and since- clauses often come at the beginning of sentences. .....As it's raining again, we'll have to stay at home. [...]
[They] are relatively formal; in an informal style, the same ideas are often expressed with so. ......It's raining again. so we'll have to stay at home.
2. because Because puts more emphasis on the reason and most often introduces new information which is not known to the listener/reader. .....Because I was ill for six months, I lost my job. When the reason is the most important part of the sentence, the because-clause usually comes at the end. It can also stand alone. Since and as cannot be used like this. .....Why am I leaving? I'm leaving because I'm fed up!
........(NOT...I'm leaving as/since I'm fed up!) .....Why are you laughing? ~ Because you look so funny.
A because-clause can be used to say how one knows something.
.....You didn't tell me the truth, because I found the money in your room. ........(= ... I know because I found ...)
Swan, Michael (1980) Practical English Usage (3rd ed, 2005.73), Oxford: OUP