As a noob (@ welding, I mean), can one of you more learned types give us a relatively simple explanation of the differences (in terms of how they affect use) between inverter and transformer welding units? What I mean is not so much an electronics lesson as a practical understanding of them.
Here's my take.
A transformer welder is just that: a transformer. It's two big coils of wire that converts 50 amp 240v coming out of your wall into 180-amps at 30 volts or thereabouts. If you have an AC/DC transformer welder, then it also has some diodes in it that "rectify" the AC to DC. And that is pretty much it. Add an on/off switch and a movable tap to control the output, and another big coil of wire called a "choke" to smooth the output, and you have the welder. Basically, it's just a big hose that takes the output from your wall and shoots it out the welding leads. Now, there are some transformer welders that implement more sophisticated features, but that's almost never what people are talking about when they're talking about budget welders. Open up a "tombstone" or "buzzbox" welder, and you will find: on/off switch, taps for output control, big honking transformer, rectifier array (diodes), and one or two other small odds and ends.
An inverter welders uses transistors to control the output at the leads. A transistor is like a tiny electronic switch that can turn electricity on and off very, very quickly (like, thousands of times per second). What this means is that an inverter can shape the electronic wave-form at the leads much, much more precisely and in much more sophisticated ways than a transformer can. A transformer is basically beholden to what comes out of the wall; an inverter can make the electricity sit up and do tricks! For example, the rectifier array used in transformer welders is supposed to turn the sine-wave of the AC that comes out of your wall into a perfect, flat DC signal, but in reality, there is some "ripple" of the AC waveform that sneaks through. As a result, your arc has some "wavering" in its output, getting slightly stronger and weaker just by its very nature. With an inverter welder, this isn't present, and the output of the welder is very, very close to true DC. This means that you get a very stable and controllable arc.
The real place where inverter welders shine is in their ability to manipulate heat output in real-time as you weld. Two features that are common are "hot start" and "arc force" (also known as "dig"). With hot start, the welder increases heat output by some percent for a second or two just after you start the arc. The issue here is that it takes more heat to establish the puddle, and the rod is more likely to stick when it is cold. This is a problem that experienced operators can overcome with practice, but even experienced operators can get into trouble in tricky situations, like an open root weld with a deep bevel, for example. Hot start helps prevent rod sticking and helps get the puddle established quickly so you can get to welding.
Arc force is a little harder to explain. When you're stick welding, the shorter your arc, the lower the heat input, and vice versa. This means that if you shorten the arc too much, it goes out and the rod sticks. With arc force, the welder detects that the arc is getting shorter, and when it gets beyond a certain threshold, the welder kicks up the output to compensate, preventing sticking. Arc force is usually sold as a way of preventing the rod from sticking when you short-arc, but really, it totally changes the way you run the rod, especially rods like 6010/6011, which are run with a "whip and pause" motion that relies on changing the arc length intentionally. One reason the SA-200 is so revered among pipeliners is that its volt/amps curve allows the welder to increase or decrease heat output over a significant range (like, maybe +/- 20 amps) by increasing or decreasing arc length. This meant that the welders could weld all the way around the pipe (flat, downhill, overhead) without ever bothering to move the output dial on the welder. Until inverters came along, plugs-into-the-wall welders didn't really have the ability to emulate this behavior, but now they do.
It should be said that, especially for beginners, having heat output change +/- 20 amps when you change arc length, is actually a bad thing. Beginners aren't steady enough to hold their arc length consistent, so they would be all over the place, much like a person would be if you stuck them in a high-end Ferarri and then turned off all the electronic stability and traction control. But the cool thing about inverter welders is that the designers can basically make them behave exactly how they want. Want a welder that puts out an output similar to an SA-200, and sell it to pipeliners? Can do. Most mass-market inverter welders keep a relatively steady heat output over a range of arc lengths, unless you really short-arc, in which case they crank up the output to keep the rod from sticking--and the more experienced user can do this intentionally to manipulate the puddle.
Although we're talking about stick and not TIG, it must be said that inverter welders have really revolutionized AC TIG. Before inverters, AC TIG was always done at 60 Hz, because that was what came out of the wall. Now, we can have pretty much any frequency we want, which changes the shape of the arc cone. Also, TIG welders now have pulse features that very advanced operators can take advantage of.
The flip-side of all this, some will argue, is reduced reliability. There are gob-loads of tombstone buzzboxes that were made in the 60's still welding today. As long as you don't exceed the duty cycle and burn it up, there really is not much to go wrong on them. Maybe replace the power switch every 25 years or so. But like James said, you're probably not using a rotary dial telephone, are you? And, frankly, we shouldn't assume that these new inverter welders won't have good service lives. There's no reason why they can't be designed to last for years and years. Do you think Shield Arc's $5000 (or thereabouts--I don't know) Lincoln V350 is going to crap out after ten years and need a new mainboard? I doubt it. But it is true that repairs on an inverter are basically always more expensive than repairs on a transformer. On a transformer welder, if the diodes go bad, it's between $25 and $100 to replace them, depending on how many you replace. I had to put a new fan in my welder once: $20. You get the picture. Meanwhile, a new board for an inverter may cost half as much as the inverter itself did, new. And repairs are seldom effected on inverters--just replace the part that's broke and move on.
For me, I don't care if my welder is going to last 30 years. Frankly, I don't have a lot of things that I had 30 years ago. In 30 years, I will probably either A) not be welding any more, and have moved on to some other fun thing, or B) sold this welder and bought a bigger, better, spiffier one. For me, an inverter is the clear right choice, because I get to, metaphorically speaking, drive a Ferarri.