Well it depends, many people see the case as part of the system which everyone is exposed to. So for that reason they choose something stylish.
That said many of the "Stylish" cases aren't necessary the best ones.
Lain Li are highly regarded but very expensive, so you can probably rule them out.
What are you using it for, Gaming rig? Media centre? Is it going to be on all the time?
Good airflow is always top of the agenda with most high power components, cases which provide this are the Antec
1200,
902 +
300 etc,
Coolermaster HAF cases are meant to be rather breezy too but I've never used one personally.
Space, big graphics cards need lots of room! Same goes for extra HDD's etc, depends on what your looking to use it for.
Tool-less installation - this is a brilliant feature if, like me, your constantly in and out the case, makes things much easier, but again it comes at a cost.
Front IO Panel - also rather handy if you don't have access to the rear.
Make sure and check to see if it's an ATX, MATX or EATX case aswell, if it's purely a MATX (mirco) it won't fit your standard ATX sized motherboard.
All in all, depends what your going to use the machine for which will make the decision of the case.
No, getting a terrible case won't technically affect the system, if you're running at stock then heat won't always be an issue,
depending what your running (dual 5970's will generate a ton of heat for example) so you could go for something cheaper.
If you are running high end gear then perhaps you might need to think about investing abit more.
Saves you buying cheap, then spending more money in the future.