What are some of the basic differences- number of buttons, difficulty in changing modes, ability to change battery,makeup of screen material, downloadability, etc, that I should be looking for and what are their relatve importance?
Since no one had really addressed these last questions while we all focused a lot on the subject of conservatism, I thought I'd start up the discussion of the other factors manni-yunk had asked about. I'd guess that the OP is more in need of a basic dive computer primer than a detailed discussion of the machinations of using tri-mix with each option. I'll start us off, although there are many on these boards who can offer better info. Here goes. There are quite a few things that are important to consider if you're a first time dive computer buyer:
Buttons: With respect to buttons, there are a couple of schools of thought. One school is that more buttons equals more usability and easier navigation, due to fewer buttons presses required to get to a specific sub-menus. (i.e. it's a little easier not to get lost and to quickly access the fields you want.) Others find that fewer buttons are easier and less confusing (a la iPod/iPhone). Personally, I've tended to find the machines with 3 or 4 buttons the easiest, since there's too much button pushing for my taste to navigate most 1 and 2 button machines. But there are definitely exceptions, depending on how cleverly their software was written. In general, at least 3 or 4 buttons may make your life easier. You should try to get your hands on a few computers and see which you like best. This will definitely affect your comfort with your computer, and you want to have one that you're not wrestling with and jumping through hoops while you're diving.
Physical configuration: Another thing to think about when buying a first computer is whether you want to wear it on your wrist or carry it on your console. This is really a personal choice, and there are excellent models of each kind. Some specific models can be configured both ways. Further, if you decide to carry it on your wrist, you'll have to decide between the convenience and smaller size/streamlining of a computer that has a wristwatch configuration and a larger wrist-mounted computer that you would never wear out of the water, but may have a larger display and bigger buttons. It's worth looking at both and seeing which is more comfortable for you. There is no right answer on this one...it's a trade-off. Many people like a smaller machine that they can keep on their wrists during their whole trip, to prevent leaving it somewhere or having to carry it in a case, etc. It's also a little easier to move underwater if it's not too bulky and getting in the way of things. On the other hand, it's important to make sure the display is totally readable and gives you clear, helpful info at a glance regardless of the visibility, etc.
Battery: Having user-changeable batteries can be really convenient. If you head out on a trip, and your battery happens to die on one of your first dives, you're stuck with tables and their inherent limitations for the rest of your trip. That can suck. When a computer doesn't have user-serviceable batteries, it can take weeks to get the machine serviced, since many have to go back to the manufacturer, not just back to your LDS. Still, there are many excellent machines that aren't user-changeable. Often, the wristwatch style computers don't allow you to change their batteries yourself due to their configuration/sealing and size. It's another case of a choice you have to make by weighing pros and cons for your needs, with no single correct answer. A lot of companies have charts on their websites that list which models have user-changeable batteries and which don't, so a little research will help in this regard. Ultimately, though, if you're someone who only dives on specific trips a few times a year, it's not that big a deal, as long as you stay ahead of the game. You just send in your computer for a new battery before the trip, and you'll have no problem. It's easy to work around the issue with a little planning, and probably isn't the determining factor in picking a computer. Further, you should be servicing your computer on a regular interval anyway, so you'll have to send it in for that at an interval fairly close to battery replacement anyway.
Downloadability: The first thing here is whether you're on a PC or a Mac. The number of dive computers that offer native Mac interfaces is limited. So if you're a Mac user and don't want to run Parallels or the like, that may determine which computers you look at. However, there are also some pretty good third party software solutions available for certain computers. For example, DiveLog works very well if you have any Suunto, and doesn't cost much. There are also freeware programs available (e.g. MacDive) that are pretty good, and support quite a few popular computers. If you're on a PC, then it's basically which company has the best interface. Pretty much every major computer brand has some degree of connectivity to a PC, and maybe others can chime in here on which ones they feel have the best software, etc.
In terms of relative importance, it is, again, personal. In my case, I'd put configuration/placement/size and the button interface ahead of the battery issue, as that can often be worked around. Comfort with your machine while you're diving is the most important thing. Still, all the above factors do matter, so you may or may not have the same priorities as I do. Downloading is also important, but it's another area that can be worked around if absolutely necessary. And since most do interface with PC's, at least to some extent, you shouldn't have too much trouble in that regard. Even though some software is better than others, I personally would want to make sure that I had a configuration that I liked first.
The machines: I thought I'd once again try to distill things a little bit, in the interest of making this thread specifically useful for the OP, as opposed to being such a general discussion of good dive computers. While many of the models being discussed here are certainly top notch machines, many of them are far more expensive than the basic models also being discussed. I think a little separation is in order for the OP and others who may be looking for a more basic first computer. For example, the Liquivision, while an amazing machine, has a list price in the $1700 neighborhood. Even the Uwatecs, which are also great computers, are mostly $1000 and up, and the more feature-rich models are in the $2000 neighborhood (although it is possible to get a basic/stripped down model for a little bit under $1000). Comparing any of these higher end machines to the likes of a Gekko or similar computer is really inappropriate, as the Gekko costs a little over $200, and isn't trying to compete with those. Further, the kind of person looking at one is unlikely to also be looking at the other. Every shopper has a price range.
Maybe if the OP tells us what he was hoping to spend, we can narrow things down a bit. (Also useful would be an overview of the OP's experience, training, and typical usage.)
Buy big or trade up later?: While it's smart to get something that will still be useful as you progress as a diver, this can sometimes also lead to wasted money. One good strategy for him might be to purchase a more basic computer to start with, albeit one that still does feature all the functions he has mentioned (Nitrox capability, etc.), and use that as a learning experience in determining what his future needs will be. There are some great computers available now that don't cost much, and there will be no loss of reliability or safety by choosing one of those. The only difference is features. In some ways, it might make more sense to spend $225 on a Gekko now, and then upgrade later if he decides he is in fact interested in tech diving or the like. All technology tends to come down in price over time, and siginificant advancements have been made in dive computers literally every year in the recent past. So by the time he completes the more advanced training required to use the tech-related features of a machine, it is very likely to be out of date before he's even used them. Therefore, it doesn't make much sense (in my eyes) to invest a large amount of money in a high-end computer right now that performs functions that he isn't going to use for several years until he goes through much more dive training (if he ever uses them at all). The chance that he "might" eventually use them doesn't really hold water, as there will be better machines for the same or lower price that do those things by the time he gets to them. And since $200-$300 isn't that much of an investment, he won't have to stress about the decision as much, and can upgrade based on his actual future needs rather than having to guess them now before spending close to $2,000. Just my two cents.
Hope this helps!