Conversations typically trim portions of messages to make it look you only see what people actually type out. It strips things like headers, signatures, replies, etc.
Threads are a sort of pre-Web 2.0 way of doing things. The messages are linked together in the tree portion of the UI but are still viewed individually in the viewing pane. They aren't stacked together like they are in a typical conversation view.
Conversations are a more modern, succinct take on what threads were trying to be.
I use gmail, and I use Outlook and PINE and Netscape Mail and Eudora etc and I can't work out what was improved.