I didn't really get the wiki explanation:
A semantic, organized place to put your server-sided game logic, which does not interfere with the world. Scripts will run inside this service, and will not replicate to game clients, allowing for secure storage of your scripts.
One reason (among others i.e language used) that I don't get it is that in the tutorial "Lighting Parts on Fire" (http://wiki.roblox.com/index.php?title=Lighting_Parts_on_Fire) the author clearly states:
We also need a Script in ServerScriptService.
But the tutorial is dealing with parts in the 'world' or Workspace I assume so doesn't this go against the wiki explanation of ServerScriptService? If someone smarter than me could just explain it in more layman's terms please :)

2 comments
What constitutes 'modifying the client'?
Character Movement, Moving a TextButton on the screen, mostly it's just messing around with GUIs. If you want to modify the client then you want to use a LocalScript