The BlackBerry is actually a smart phone. One of the differences between a smart phone and a PDA is the inclusion of phone capabilities with the smart phone. The smart phone market has really taken off because people want that all in one mobile device. And with the ubiquitous nature of mobile phones, it just made sense to combine the two (PDA-type functions with a mobile phone).
(Jeremy great answer...) The one other difference that my colleague might have mentioned is that Blackberry uses PUSH technology, the mail is pushed to the device with you having to do anything. Treo and other Windows Mobile devices (so far) are PULL technology, you have to tell the device to go get your mail - you can set it up to get it every so often, whereas Blackberry immediately sends mail to your device as it comes in.