The Differences Between Dual Mode & GSM
Throughout the world, other than North America, the standard for telephone data transfers is GSM. In the US, the dominant standard is CDMA, and more and more data phones are built around being able to access WiFi networks. Any telephone that can handle more than one mechanism for transferring data to and from the phone is a "dual mode" or "multi-mode" phone. Dual mode phones are often (but not always) multi-band phones as well, with antennae that allow them to switch between frequencies based on their locations.
Dual Wireless Standard Phones
Dual wireless standard phones, sometimes called "world phones" have both CDMA and GSM radios in them. These effectively function as two phones in one device, and modern ones will turn the radio not in use to standby mode automatically. It is also possible to turn off the data transfer technology that isn't being used to save power, or to avoid going over the download limits of a particular data plan. By contrast, a GSM phone can only transfer data and work as an Internet-enabled device when a GSM network is present.
Wireless Data & WiFi
More phones are built to accommodate one of the two wireless data standards (GSM or CDMA) and have an 802.11 wireless antenna, just like a laptop computer does. This is known as "data plan plus WiFi". These phones have the advantage of being able to go "off the data plan" whenever a WiFi connection is available, which can greatly reduce the amount of data consumed, and thus reduce the monthly data plan bill (if it's not an unlimited plan). Most pure GSM phones do not have the ability to connect to a WiFi hot spot.
The primary benefit of using a GSM phone is that it can work in most countries; the primary benefit of a dual mode phone is that dual mode phones give an expanded coverage area within a given country, or allow you to use the Internet features of your phone on a WiFi network without consuming units of your data plan. The most common variety of dual mode phone on the market is a GSM+WiFi phone.
Drawbacks of Dual Mode Phones
Dual mode phones have two radios, and those radios consume electricity. Knowing when to turn off the other radio to save battery power is one of the things that veteran smart phone users learn. For example, if you're not going to be in a place where a WiFi signal is likely, turning off the WiFi can add some time to the battery life on the phone.