I get various hints at this question and I figured I would try to explain the technical details with more rigor. This has been a goal of mine for some time now, and honestly the main benefit can be expressed as 'liberation'. That is, having Elliquiy on a dedicated platform means that I have options that simply weren't available to me before.
The first of these is probably the most obvious during the transition - http://70.32.39.16 takes you straight to Elliquiy, no ifs, ands, buts (butts?), or whatnot. Sure playing around can take some more effort, but at least if your DNS goes down you can bookmark a link that will take you straight to Elliquiy even if your ISP fails you.
This has a lot of little site benefits. One of these is reverse DNS - I can have 70.32.39.16 point to elliquiy.com, just as elliquiy.com (now) points to 70.32.39.16. This may seem silly and geeky, but remember all the problems we had with Yahoo thinking we were a spammer? Granted, I'm crossing my fingers, here, but that sort of matching helps get past spam filters, because now e-mail that comes from elliquiy.com will look like it's really coming from elliquiy.com, no way we're not kidding, we mean it.
A third benefit the dedicated IP provides is 'real' SSL, which is difficult to have multiple sites targeting the same ip address with as on a shared host. After the dns problems get properly resolved, I should be able to get https://elliquiy.com working right, meaning you more paranoid types can enjoy that much more privacy.
That's just the cozy benefits we get from having our own IP, of course. Having full control of the server proper means I can install what I want on it. Not just fun little projects that I don't want to clutter Elliquiy proper with, but things like Sphinx search, which is basically a dedicated search program that works outside of the limitations of php itself, and not only is it significantly faster, it can handle stemming - a search for 'like' also looks for 'likely' and so on. Sphinx handles Russian stemming, too >_>. The main cost here is that Sphinx is optimized for search speed and indexing is a pain, but I imagine most people are more interested in what they wrote last week as opposed to an hour ago.
And, of course, we get our own guaranteed piece of the server pie. It was possible for Elliquiy at points to be using over half a gig of RAM, and on a nightly basis we saturate the processor, which can have less than fun results when the other major site the previous server hosted decided to have itself a good helping of blueberry. This isn't perfect, of course, but assuming donations continue like they have we can buy a more solid position : )
...all of this, of course, doesn't really mitigate Elliquiy's primary problem (besides having me at the helm) which most members can at least agree on the general nature of. SMF 2, the next version of the forum software that Elliquiy currently runs on, makes permission handling a great deal easier, so large, private groups can exist within Elliquiy (and forums properly ignored) without the goddesses and I having to manually check for and grant each individual request. I don't presume to think that will make everyone happy, but at least I can make the forums seem 'less huge' which is a major problem right now. Especially as the stress of economic events tears on people, and snapping occurs before understanding - something I'm rather guilty of myself.
I plan on upgrading when it goes into release candidate mode, which I'm hoping to occur before the year's end. Fingers are crossed there, too.
Ultimately this, Drupal, upgrading forum software, getting the Wiki more attention... are all stopgap measures. I've mentioned before that I have much bigger plans, and I really do think that social problems can have technical solutions. Hopefully you will all bear with me for the time it takes me to begin to prove my case.
I love you all, I really do. Yes, you. Especially you. Maybe not in that way, especially not if you're male... you know what I mean. -_-
P.S. We were also using 2.95 gigabytes out of 3 gigabytes. We have 60 gigabytes now >_>