While the four existing PSP models have--with the exception of the PSPgo--basically the same form factor, and the changes inside haven't been too drastic, there have been some important differences. And with the upcoming release of the PSP's successor, PS Vita (code-named NGP or Next Generation Portable), and the recent appearance of the Xperia Play smartphone (aka "PSP Phone")the changes get a bit bigger. Here's a rundown of the four PSPs and the PS Vita, with links to detailed lists of specs.
It seems a bit heavy and clunky now, but when the PSP first came out, it was sleek and shiny and powerful. The screen is bright enough and big enough to make watching movies a great on-the-go experience, and if the games aren't as graphically detailed as their full-size-console cousins, they were still miles better than the competition. The original PSP was envisioned as a multi-media device, with the hardware to handle movies, music, photos, and (of course) games.
The second PSP model was dubbed the "PSP Slim" (or "PSP Slim and Lite" in Europe) by fans, because it significantly reduced the thickness and weight of the device. Hardware changes were fairly minimal, but included an improved screen, a better UMD door, and a faster processor. To make the thin silhouette, a few switches were moved around. An addition to the firmware that was PSP-2000 only (at the time) gave users Skype, so the PSP could even be used as a phone.
The main change to the third PSP model (aside from a somewhat improved battery) was the brighter LCD screen, leading to its nickname, "PSP Brite." Early on some users claimed to be able to see scan lines on the screen, leading to many people deciding to stick with the earlier 2000 model. There don't seem to be problems with the screen anymore, and the PSP-3000 is generally considered to be the best of the four PSPs (unless you're a hardcore homebrewer, in which case the PSP-1000 is preferred for the ability to downgrade the firmware).
The PSPgo is fairly obviously different from its siblings, though it's primarily cosmetic. Aside from the complete lack of UMD drive, it functions much the same as the PSP-3000, but in a smaller, more portable size..
The PSP-E1000 (which doesn't have a nickname yet, but I'd like to suggest "PSP Extra-lite") was a bit of a surprise announcement at Sony's 2011 Gamescom press conference. So far only announced for Europe, the PSP-E1000 features a minor cosmetic redesign, and loses the WiFi featured in other models. It also has mono instead of stereo sound and a slightly smaller screen than the other PSP models (not counting the PSPgo).
From the looks of things the PS Vita might be as big a deal--or even bigger--than the original PSP when it came out. Without increasing the size too drastically, the designers at Sony have added a bigger, brighter, higher-resolution screen, and considerably more powerful innards to their next portable. It's hard to say how this will translate in actual use (but I've got some ideas), but smoother, better-looking games are almost guaranteed. Backwards-compatibility, at least for downloadable games, has been promised too.
While it's not technically a PSP, the Sony Ericsson Xperia Play smartphone does have some PSP-like features, including a slide-out gamepad very much like that of the PSPgo except with touchpads instead of analog nubs.