From a customer's point of view no it should not. Linux can run ASP via Chillisoft and now .NET with mono (for the time being at least). Windows can run Apache, PHP/Perl, and MySQL so for technological offerings they are about equal.
A windows box can be secured and run properly by a competent admin. A linux box can be secured and run properly by a competent admin. Since a customer will assume the hosting company has properly secured the servers that comparison is a wash too.
Linux has no license cost for unsupported distributions, true. But since more than license costs go into pricing your service offering and customers cannot divine what percentage of costs the licensing amounts thats shouldnt be a factor.
Unless the customer has a particular affinity for either platform they both will serve up web pages and perform the various value added functions that go into web hosting. So from a service offering/security/aggregate pricing standpoint the platforms are identical (uh oh... here come the fan boys ).
If you are looking for all the windows related features like support for .asp, .net and mssql etc then I suggest you to go for Windows hosting.
But generally windows hosting is more expensive as compared to Linux hosting.
On the other hand nowadays we get .asp support (as good as the one that we get on windows) on linux with chillisoft.
But in both the cases one main thing which matters a lot is which hosting company are you choosing.
A good hosting company will give you stable servers.. !
I personally favour linux because of its domination over windows w.r.t various features available for hosting.
You should go for windows unless you need the perfect windows features ..!
Lots of hosts prefer linux hosting to run their businesses but that by no means imply that windows or coldfusion is bad, just depends on which platform you are more familiar with and would be able to handle better. thanks