Hi. I try to explain the difference between these two words.
When we say 食べ物、 飲み物 isn’t included. But when you say 食品、it means both 食べ物 and 飲み物.
And another point, I guess, is whether the material has been already cooked or not. (My English is not good enough. Sorry.)
For example wheat is referred to as 食品、but pancakes can be categorized as 食べ物.
Hopefully I can be of any help to you. Atsuko