Microsoft (for better or worse) created the MsBuild system which a project file (.csproj, .vbproj, etc) are scripts for. These .proj files direct the MsBuild system to build the project via the XML. This is essentially what nAnt is doing as well. So, generaly, MsBuild functionality == nAnt functionality.
While I've certainly spend quite a lot more time with MsBuild vs nAnt so I can't be a truly expert opinion, from my experience, if you are using Visual Studio, it is probably a better track to go down by staying with MsBuild for building your projects since it's most likely already in tune with your development process. Since you almost certainly are already using .proj files (i.e. .csproj if you are doing C#) it doesn't make much sense to me to create another build file: just use the one you already have and customize it as you see fit. I'm unaware of anything that nAnt offers at this point that MsBuild does not except for the fact that it is more in alignment with the Open Source mode.
Again, since my experience with nAnt is limited, I can't say that this is still true, but when I last used nAnt, it was quite a bit more verbose to accomplish the same tasks as compared to MsBuild, which, if still true, would be another reason to me to use MsBuild instead.
Your need to drop binaries as a post-build task is a common one easliy accomplished with MsBuild or nAnt. For the aformentioned reasons, I'd suggest looking to MsBuild first to see how this would work then looking at nAnt only if MsBuild doesn't fit your bill. I say this just because as mentioned, you are already soaking in MsBuild today (the MsBuild engine is already installed on your computer too) so why not take the path of least resistence? Certainly if MsBuild doesn't do what you want and nAnt does, by all means take that route. But copying files would be 5 lines or so of xml in your existing MsBuild (i.e. .csproj) file. Here is some documentation from MSDN: http://msdn.microsoft.com/en-us/library/3e54c37h.aspx
A previous poster mentioned using CruiseControl with nAnt. Cruise control can easily use MsBuild too:
http://ccnetlive.thoughtworks.com/ccnet/doc/CCNET/MsBuild%20Task.html
so if you want automated (Continuous Integration) builds, you can use CC.NET. MS Team Foundation Server is a alternaive to CC.NET if you can stomach the cost or are a startup (via MS BizSpark program you can use it free for a few years I think).
I don't have anything against nAnt--it's just that with what you mentioned you want to do it doesn't seem like you need it. MsBuild can do it and you don't need to install anything new or even create a file. You just need to do a 5-line (roughly) xml entry in your .proj file