The difference is that (( is a “compound command”, similar to [[ (evaluate conditional expression), while $(( )) is “aritmetic expansion”. They behave in almost exactly the same way but are used in different contexts - the former uses “exit codes” while the latter returns a string, so the former would be used where you would expect a command, while the latter would be used where you expect an expression.

This is similar to how there is ( compound command (run in a subshell), and $( ) (command substitution). You can actually use the former to define a function too (as it’s a compound command):

real_exit() { exit 1; }
fake_exit() ( exit 1 )

Calling real_exit will exit from the shell, while calling fake_exit will do nothing as the exit 1 command is executed in a separate subshell. Notice how you can also do the same in a command substition (because it runs in a subshell):

echo $(echo foo; exit 1)

Will run successfully and output foo.

It is another one of those unknown, very rarely useful features of bash.