I read at one point that one is a fan of TOS while the other is a fan of TNG. This may be a matter of opinion but I'm wondering what the consensus is, if any?
Trekkies is what Star Trek fans called themselves. The outside world only saw the most fanatical and unhinged, so that is the image that they had for Trekkies and used it to refer to all Star Trek fans.
Trekkers is a term coined by those who take themselves and what others think of them far too seriously. The did it to distance themselves from that image.
I was there for the whole, "I'm not a Trekkie, I'm a Trekker" movement. So, I know who they are. Most of them are, in fact, Trekkies. I've been using this username long enough that they know who I am. I wonder if any of them are here....
There are many Star Trek fans who just don't give a hoot about what you call us.
As a member of the former STAI (Star Trek Association of Irvine) which was started on the 10 year anniversary of the cancellation of TOS. I was there for most of it.
BTW, @KyleJones, Shatner caught a lot of blow back for that statement. It really limited his popularity at cons for a while regardless of how true that whole skit was (I've met almost everyone that was portrayed in that audience). It was worse than Nimoy's I am Not Spock book; mostly because Nimoy really didn't care.
EDIT: I just found a page that links to RAPA our (STAI's) old monthly newsletter. I didn't know those still existed.
There isn't a consensus. It used to be that trekkies were the sort of people Shatner poked fun at in the infamous "Get a life!" Saturday Night Live sketch, and trekkers were the vast majority of the audience who enjoyed the programs but didn't make a fetish of them. Since then, various stars of the programs have weighed in, Star Trek fandom has vastly expanded and the situation has become hopelessly muddled.
A (german) trekker told me about 30 years ago (i.e. when TNG was still new), that trekkie was a derisive nickname used by non-trekkers, while trekker was the name preferred and used by the trekkers.
The section "Trekkie vs. Trekker" in the Wikipedia article "Trekkie" tells the same story.
Back in the 60s and 70s, we were Trekkies. Trekkies were passionate and goofy. We didn't take ourselves all that seriously, but any time networks tried to take Trek off the air (including syndication) we mobilized and wrote letters to keep it on. Perhaps there was always a self-important segment of Trek fans, but they became more prominent in the 80s and beyond. They wanted to disassociate themselves from those weird Trekkies, so they adopted the moniker "Trekkers" and ret-conned all sorts of explanations for the distinction. However, what was born out of disdain was passed on to fans who weren't even born when the division arose. Thus, later "Trekkers" probably aren't motivated by snob-appeal--it's just the term they heard when they came into the fandom.