The difference stems from the distinction between logical laws and laws of nature. A sentence can be said to be logically necessary just in case it's true in all worlds that satisfy the logical laws, that is, in all possible worlds. A sentence can be said to be causally or physically necessary just in case it's true in all worlds that satisfy those laws of nature.

As regards the direction of implication between logical and causal necessity: what is logically necessary is true in all possible worlds, so it's true in all physically possible worlds and is thus also causally or physically necessary.

The converse, however, is not true: it's not the case that what is physically or causally necessary is also logically necessary. What is true is that if we let N be the set of laws of nature, and S a causally or physically necessary sentence, then the conditional (N [math]\rightarrow[/math] S) will be logically necessary.