• [ is a command (basically a variant of the test command). [[ is a builtin in many shells.
• When you write foo* inside [...], filename expansion (aka globbing) occurs; while inside [[...]] pattern matching occurs.
• [[ is more powerful and capable than [.
• [ and [[ are not part of the if syntax, you can use them as in [ "$exit" = 'yes' ] && exit.
• Inside [...] you should prefer = instead of ==. As far as I know, the second one is accepted in many shells but is not POSIX-compliant.

By the way, I recommend you to double-quote your variables even if you really know how word splitting will behave.

• [ is a command (basically a variant of the test command). [[ is a builtin in many shells.
• When you write foo* inside [...] filename expansion (aka globbing) occurs; while inside [[...]] pattern matching occurs.
• [[ is more powerful and capable than [ but should not be used if portability is a concern.
• [ and [[ are not part of the if syntax, you can use them as in [ "$exit" = 'yes' ] && exit.
• Inside [...] you should prefer = instead of ==. As far as I know, the second one is accepted in many shells but is not POSIX-compliant.

By the way, I recommend you to double-quote your variables even if you really know how word splitting will behave.