I think there are some situations where you could use either, and there are some situations where only one is right. The below examples are taken from the Core 6000 set.
When telling someone to do something, or promising someone you are going to do something, I think 必ず is the right thing to use.
この薬を必ず飲んでください。 — Be sure to take this medicine.
この借りは必ず返します。 — I will definitely pay it back.
On the other hand, when saying “more than usual” you use いつも.
彼女はいつもより早く眠りに着いたよ。 — She fell asleep quicker than usual.
If the meaning is just “always”, there are examples where both are used:
朝ご飯にはいつもパンを食べる。 — I always eat bread for breakfast
うちでは、朝食には必ず梅干しが出ます。 — In my family, pickled plums are always served for breakfast.