WordPress uses cookie based authentication. When you log into wordpress using curl, wordpress is authenticating the server(which ran the curl script) & not your client. so when you visit wordpress using your browser(client), it’s not authenticated.
For the solution, you could use wordpress hooks to automatically authenticate the user to your site just after they authenticate themselves to wordpress. Then whenever a user comes to your site, you could just redirect them to wordpress login url for authentication.
Another solution is to override the authentication functions in wordpress. All of the authentication functions in wordpress are directly or indirectly pluggable. When you override them instead of checking if the user has the correct cookie set, you will check if the user has logged in on the other site (Maybe use a shared database?)