Well if you are using a torrent program you are not downloading data from a website you are downloading it from a peer (or more usually many peers). But that is neither here nor there.
What kind of site is it?
Is it a private site or is it a public site (like The Pirate Bay)?
If the site is a true private site the fact that you can download with another torrent client is usefull information. Just to be clear sites like demonoid are not
private sites Demonoid is an invite only public site. If the site is a public site . . . it is compeletly irravelent that you can download with another client.
The reason I ask is that those errors usually indicate a problem with the tracker. However, since public sites allow any number of trackers in their .torrent files . . . it might just be one tracker which is experiening problems which for some random reason all of your other torrents are using. But the .torrent file you downloaded and opened with uTorrent might have a differnet tracker or set of trackers or is might just not have the problem tracker in it.
Once you sucessfully downloaded the data using uTorrent did you check if you were still getting those errors in Vuze? It is possible (but unlikely) that the torrent-tracker problem was solved while you were downloading uTorrent and grabbing the .torrent file which worked correctly.
It could be a wierd traffic shaping problem. Have you checked to see which port Vuze is using and which port(s) uTorrent are using? If for example you are using a standard port for Vuze which your ISP is blocking it is vaguly possible that could cause that error (but I seriousl doubt it) it only takes a few minutes to check. You should be using a port in the range of roughly 50,000 - 65,000. If the port you are using is different . . . try changing it to a port in that range . . . it is not going to hurt anything. If you have manually forwarded your torrenting ports of course you will need to change your port forwarding. But if you have done it once it is pretty easy to change it to a different port.
If chaning the port helps . . . you might be being traffic shaped . . . so take a look at this:
first check here
and locate your ISP.
then read this
for suggestions on how to avoid your ISP's efforts (if necessary)
The words here and this are links ;-)
Hmm that is all I can think of for now. Give all that a try and let us know a little more info!