Hello and welcome! Sorry, it took so long for someone to respond. The WAP browser which is a mobile browser found on most cell phones. It is a scaled down browser, only able to access certain websites designed for mobile phone use. The other one, the Blackberry browser, is a full-featured HTML browser (like the ones found on a PC, i.e. Internet Explorer, Firefox, Chrome, etc.) which can access most all websites (not just mobile sites) and is the better browser that you really would want to use primarily instead of the WAP browser. The full HTML browser is one of the features of a BB (and other smartphones and some recent higher end regular phones) that normal cellphones don't have and is one of the reasons these devices are used by people who need frequent and full mobile access to the net.
The experiences on a full HTML Browser (non-BB) and that of a BB with a similar full HTML Browser are now essentially the same.
Well hold on . . . the regular BB Curve has only a WAP Browser!
The Touch has was is called a "full html browser"
The Env3 has was is called an "html browser"
The BB Curve has was is called an "Blackberry Browser"
Now they are all forms of the WAP style but differences are in how rich an experience they are capable of rendering. This gets really complicated. I've cut and pasted the below to try and explain the differences.
WAP / Web Browser
Also known as "Wireless Internet", "Wireless Web", or "Mobile Web". This feature means that the phone contains mini-browser software that will let you navigate various services through the Internet, shown on the phone's display.
Most older phones with this feature can only display "WAP" pages specifically designed for phones. Many newer phones can also display full web sites, although how well the page fits on the display varies depending on the browser software.
Some traditional web sites do offer versions specifically designed for phones, however frequently only a fraction of the content or services are offered in the wireless version, such as features that are time- or location-sensitive.
Several different standards exists: HDML, WML, cHTML, and xHTML. Older phones support HDML and WML. Most new phones support WML and xHTML.
xHTML, which is related to WAP 2.0, is the standard that the industry is converging on for mobile-optimized web sites, although phones designed to browse the "full web" are increasingly popular.
WAP 2.0 and the full web can be accessed over any type of data connection that reaches the Internet. The cell phone network is most commonly used from phones (with 3G and 4G providing faster service) but short-range options such as Wi-Fi can be used as well, if the phone supports it.
Confusing is right! You can say that again. Part of the problem here is each manufacturer seems to want to intentionally use all kinds of different alphabet soup verbiage to describe their browser and somehow make it look like it's something super unique. So let's make this simple if we can. It took me a while to get back to you on this but what I have found are the following top Verizon models to check out for the fullest HTML Browser experience:
LG Env Touch
Samsung Omnia
Blackberry Storm
HTC Touch Diamond
HTC Touch Pro
These will give you the richest fullest internet experience you can find right now. Like I said, they all describe their browser slightly different from the next so that it's nearly impossible to see the same verbiage. I would look at all of them and make a decision about which device provides the richest Internet experience to YOUR eye. Note: the only one not requireing the more expensive data plan is the Env Touch 
