Not all users will care but the important thing is to make all (well as many as possible) of your visitors feel comfortable.
So if your site is about a network (of sites/people etc) use .net if its non commercial/community focused/[or a soft sell ;-)] use .org
.Org is an unrestricted gTLD, just like .net, marketed as the domain of NPOs and various other organizations: community, industry, social, etc.
You can pick any gTLD you qualify for but, to the best of my grok-ing, whether anyone choses to engage your website will have a great deal more to do with the website itself than whatever gTLD you choose.
One can easily build a bunch of nothing or nonsense on a .Org just as easily as they can on a .Com or .Net or .Info. I'd only make the gTLD central to website development IF that was part of a larger plan to build something particuarly fitting to the gTLD. In the case of .Org that would likely be a website light on "doing business", i.e., one without multiple AdSense blocks and banner ads pasted on every page. No matter where I come across that experience I tend to flee and not leave money - mine or any advertiser's - behind.
So it really depends on what you are trying to accomplish with the site. The .NET could be more appropriate if you are doing a more social site, or technical guide site, etc...
Just my opinion!