Blackberry's use the companys servers to get your messages on and off of your device in seconds.
With the iPhone you have to "pull" messages, meaning you have to have a set time (every 15 mins) to check for messages, or you can manually pull them whenever you want, but it can take some time, and not always work.
Blackberry messages are all stored on the device, and you can see them all offline.
iPhone messages are often not stored on the device, and have to reload again and again (you need radio access).
I made the switch from a blackberry, and while I have not turned back, I would gladly pay blackberry a monthly fee in order have their technology licensed on my phone (I am hoping for a scenerio like this). Other companies have been completely dodging push e-mail because of patent claims by several companies including BB parent RIM.
If you are using it for work, get a blackberry. I had a mobile phone and a blackberry with data only before. When I got the iphone I consolodated. I still have the blackberry, and have been toying with the idea of getting a data only connection for it and pushing work e-mail again (I actually gave up on that before the iphone b/c I starting getting ~200 messages a day thru work, and practically live in my office, so I figured there was no point...). I would never want to use an iPhone with the amount of messages I get through work, but for personal use, it is perfect and beautiful.
The internet is absolutely a no contest beatdown. I take my laptop out of the house 75% less since I have gotten the iphone. I am about ready to cancel my mobile data card because the iphone can get most things done. I am a big iphone update away from not needing the laptop for mobile internet. As soon as I can DL/UL files, etc.