Law of Detachment means 1. if p then q and 2. p then we can derive a 3rd true statement 3. q
Law of Syllogism means that if the following two statements are true 1. if p then q. 2. if q then r. Then we can derive a third true statement 3. if p then r.
So the difference is one has only p and q and the other has p q and r.