BitTyrant uses a different type of torrent protocol that's compatible with BitTorrent and is optimized for fast download performance. Now, the big selling point of this client is speed although this cannot be guaranteed depending on what type of connection you have, how many seeds a torrent has etc.
However, the developers claim that it's up to three times quicker than Azureus and much fairer in the way it rewards uploaders rather than downloaders. In terms of interface however, there's little difference - BitTyrant looks exactly like Azureus although there are a few optional features to display statistics relevant to BitTyrant’s operation. There is however an underlying difference between it and other clients. BitTyrant differs in its selection of which peers to unchoke and send rates to unchoked peers. In this way, it theoretically means that bandwidth speeds are used more efficiently and fairly although as I say, much will depend on your own connection and ISP provider.
If you're happy with Azureus then there's no big reason to change to BitTyrant but if you've been experiencing declining download speeds or connection issues, you may be pleasantly surprised.