BECAUSE (c)

      I was worried because Tally was late

In conversation we very often use the form 'CAUSE (also spelled: coz, cos or cuz, cus in American English)
      I left cause I was getting bored

BECAUSE  OF+ noun (p)

This is a preposition, so it is not followed by a clause with a verb, it is followed by a noun or noun phrase

      She was late because of the traffic
      I was tired because of all the hard work
 
AS / SINCE (c)

These conjunctions are formal. In a less formal language we prefer the equivalent with SO:

     As it's raining again, we'll have to stay at home (= It was raining again, so we had to stay at home)
     Since it's too late, I think we should stop now  (= it was too late, so I think we should stop)
 
They're used when the reason is already well known, or it is less important than the rest of the sentence.  These clauses often begin the sentence:
 
      Since I speak no German, we had to communicate in English
      Since it's too late, we should leave it for tomorrow
      It's very noisy here, since we live close to a motorway

Compare:

- Sorry, I must go because I have a meeting at 5  (I'm giving you new information)
- Since today is Sunday, let's stay at home and rest  (we both know that today is Sunday)

THAT'S (THE REASON) WHY (c.)

This relative clause is very commonly used to express reason (especially in its short version)
 
     My car's broken, that's the reason why I came walking
     I didn’t know she was your sister, that’s why I didn't talk to her
     You speak very good English. Oh yes, I forgot your mother's English. That's why

abbreviations:  (c.)= conjunction  (p.)= preposition    (see the important difference here: Kinds of Connectors)