I keep hearing these terms being thrown around at conferences, but I'm not really sure what they really mean, if there is any difference between them.
"Research intensive" and "research extensive" are obsolete terms from the Carnegie classification of research universities. They are terrible, confusing terms that should never have been introduced (see page 5 of Rethinking and reframing the Carnegie classification). The idea was that "research extensive" means there's an extensive research program covering many areas at a high level, which "research intensive" means there's just a narrowly focused, intensive research program in certain areas. Basically, extensive is supposed to be better than intensive. Of course the problem is that this really isn't what they sound like. For example, most people would say Harvard is a research intensive university, but in this classification it's not. Because of massive confusion, the Carnegie classification was updated to use other terms, but the old terms still persist. One reason is that some people just got used to them and found it hard to develop new habits; another is that every time the classification changes, some universities end up moving to a less prestigious category than they used to be in, so they have a strong incentive to describe themselves using the old terms.