SSDs are primarily faster because they don't require any time to 'find' data. Much like RAM, any piece of data is accessible nearly instantly; versus a traditional HDD which must spin up and find any given piece of data on the disk.
The downside, is obviously the $/GB you pay for that extra speed. Typically it's most convenient to have a large HDD to store stuff locally; I wouldn't hesitate to steal the HDD from your other PC (assuming it's your old one) and put it in the new build to pair with the SSD.
Not everything benefits from the speed SSDs offer. They are very nice to boot windows from (your computer will turn on and be usable in about 20sec), and they will reduce the level loading times in games by a lot (for games that have long load times, like RPGs, this can be very very nice); however, most multiplayer games don't really see any benefit because you still have to wait for the server and the other players to load typically before a game begins.