A file uploaded with the extension seems to receive the ASP.NET file user rights.
By "the extension" do you mean NeatUpload or does the problem only occur with filenames that have a certain extension (which you accidentally ommitted in your post)?
This makes it impossible for an anonymous user to download the file as it has insufficient rights.
In the InputFile.MoveTo method there's no way to set these rights.
Can you set the rights after you move it?
I can't find any documentation for changing file rights from .NET at the moment. I'm not sure if I'm being brain-dead (it's late...) or it just can't be done from managed code. I have a vague recollection that it can't be done from managed code (at least not it 1.1), but I'm not sure...