Difference between (bash) shell variables which are exported and those which are not

Bash seems to differentiate between variables which are exported and those which are not. example: $ FOO=BAR $ set | grep FOO FOO=BAR $ env | grep FOO set sees the variable but env does not. $ export BAR=FOO $ set | grep FOO BAR=FOO FOO=BAR $ env | grep FOO BAR=FOO set sees both variables but env sees only the exported variable. I know that set is a bash builtin and env is not. What are the differences between variables which are exported and those which are not?