Lise's definition covers it, but:
A Blackberry is a smartphone, but not all smartphones are Blackberrys. Blackberry is a specific piece of equipment, or the technology/software/service related to the company who makes that equipment.
You can have smartphones that do what Blackberry does without being a Blackberry. There are different ways to send and retrieve e-mail, and some companies are making a killing with their Blackberry charges.
T-Mobile operates its own networks, which were set up when T-Mobile was Deutsche Telekom, and gives you an option between the Blackberry system and their own.
Here's the real killer: in order to take advantage of all of this crap, you have to sign on to two plans. In otherwords, it's like having two bloody phones, because you have to choose a plan to talk to people, and a plan to use the e-mail and web and other fun stuff. The companies justify this by coyly suggesting that you are going to use this thing instead of your computer, and therefore it's just like paying for your online access in addition to the phone service... .... ..... ....
Thieving bastards.
The reality is that all phone companies realize that it is becoming increasingly impossible to get money for actually talking to people, so they are taking it out of the data services. (Yes, you can now use Skype to make calls on some smartphones,)
But why not simply admit that the pay-to-talk era is over, and just charge people what you would to use your general services? What seems disingenuous is the "double-dip while we can get it" approach that all companies are currently taking.
/rant