Western Armenian is spoken by the descendants of those who lived in what is today Eastern Turkey and survived the Genocide. It's basically spoken in diaspora, which is why some people talk about it eventually dying out. I think Isa is right, though, that it should survive at least in Syria and Lebanon. Armenians living in the Middle East (except Iran) speak Western Armenian.
I think what happened was that the two dialects came into existence because of the division of the Armenian homeland between the Ottomans and the Persians. The Armenians under the Ottomans developed the Western dialect and those under the Persians developed the Eastern dialect. Around the early 1800's part of the land that was under the Persians came to be taken by the Russians, and that is the Republic of Armenia today. So Western Armenian was spoken by the Armenians under the Turks, while Eastern Armenian was spoken by those who were with the Persians and Russians.
There are significant differences between the two dialects. Phonologically, Eastern Armenian preserved the Classical Armenian distinction between aspirated and unaspirated stops and affricates, while Western Armenian dropped the distinction. What was an unaspirated p in Classical Armenian became a b in Modern Western Armenian. The b in Classical Armenian became an aspirated p in Western Armenian. The aspirated p in Classical Armenian remained an aspirated p in Western Armenian. Eastern Armenian, however, retained all that unchanged. The same thing happened with t/d, k/g, ch/j, and ts/dz.
You can see this in names from the Bible. For example, "Peter" in Classical Armenian and Eastern Armenian is "Petros." In Western Armenian, it is "Bedros." It results in words from the other dialect just sounding weird to the ear.
There are also differences in vocabulary. Even ordinary words can be different. "Jermag" is "white" in Western Armenian, while in Eastern Armenian it is "speedag." (Or is it "speetak?")
Probably the most confusing thing, though, is that the grammar is also different. Verbs are conjugated differently, and nouns are to some extent declined differently. Western Armenian has six grammatical cases, while Eastern has seven. (Eastern Armenian has a locative case. I kind of wish we had that in Western Armenian. It seems useful.
Some people can understand both dialects, and even speak both dialects, effortlessly. Others have a lot of trouble. I can understand Eastern if the person slows down for me and speaks simply. My mom doesn't understand Eastern at all, and Armenian was her first language. She didn't come across Eastern Armenian until the late 1970's when Armenians started coming here from Iran. It was just too hard for her to adjust to.