Is your other website in the same server?
You can include wp-load.php to your other website (located in the root of your wordpress installation) and get the content from there. Of course you’ll have to include stuff like the javascript libraries and the css used in wordpress (mainly those used in front end).
For different servers, you can get the data via RSS but only text content and images pointed inside RSS. You could also make pages that return you a jsonp object and use javascript on the other website to render your content.