Restarting your machine might have stopped the connection to the server and you need to start the server again. Or set it to run when windows starts, typically done by setting it to run as a service in windows.
It's rare that restarting a computer would cause you to get a new IP. Restarting a Router might. But a Computer, wouldn't. The only case that springs to mind, is if you have a DSL Modem Card, then your ISP might assign you another IP
Why don't you restart your machine and see if your IP actually changes?
Are you sure that your IP changed?
You should check your IP while connected, and after restarting your computer (run the IPCONFIG command from the cmd prompt). See if it changes. If it doesn't, then this problem has nothing to do wit dynamic and static IPs.
There are servies others have mentioned like DYNDNS, where you don't get a static IP, but you get a Domain name, and that points to your dynamic IP. Your computer or your router have a feature or software that send your IP to the dyndns people and the IP associated with the domain gets updated to your latest IP, and people just access the domain(which doesn't change).