If you want to dive, cross into Egypt and go to Dahab. Don't waste your time in Eilat, it's boring, expensive and the reef has been damaged.
Worth noting that Jerusalem is the heart and soul of both Israel and the Palestinian territory. You will quickly discover that while West Jerusalem is Israeli, East Jerusalem is Palestinian. The city has a foot in both worlds, with each side having its own language and its own culture. That's one of the reasons why the place is so fascinating: you can change country just by crossing a street.
The only place where both worlds merge is the Old City, which has been divived into several quarters depending on religion for centuries: Jews, Christians, Muslims, Armenians. There's even a tiny African quarter where you will find the descendants of Subsaharan pilgrims and merchants who decided to stay in Jerusalem.

