The main difference is the bars serve alcohol for all European games whereas it is barred in other areas. Supposedly more leg room and supposedly better quality food available. Basically if you're in the other areas of the stadium, you can buy a hotdog. In Club Wembley, a hot dog is called a "gourmet baguette" and so demands a higher price.
The Club Wembley seats extend almost to the corner flags, so the supposed 'enhanced' view is not as grand as they would have you believe. For example, I was in Club Wembley for the semi final and diagonally opposite the far side from where Eriksen crossed. From club Wembley, we all thought it had gone straight in and had no idea Kane had the final touch.
My advice is keep your money and go for the cheaper options.