The terms idioms and expressions are synonyms. This means that they are interchangeable, and have related meanings, but in different contexts they have some differences.
What are Idioms?
Idioms are a grammatical term for a group of words that only have a specific meaning in that combination.
For example: miss the boat (being late), bent out of shape (getting upset), the last straw (the final provocation), etc.
These word combinations have a specific meaning that is not dictated by the meanings of the words alone. In a grammatical sense they are one phrase, and as such are called idioms.
What are Expressions?
Expressions are defined commonly by culture. An expression is an entire sentence that has a specific meaning with that word combination. It can have the same, or similar, meaning across multiple cultures.
For example: Don’t look a gift horse in mouth. The early bird catches the worm. Etc.
Expressions might also be specific to a certain culture, and cannot be transferred. In a grammatical sense they are expanded idioms, since they follow the same rules.
In the end, idioms are linked to grammar, and expressions to culture. While they might be extremely similar, the ways in which we use them, and were we use them and how, are different.