That right there is one tremendous difference in how the game is played out. While HE focused on detectivework and managing your resources carefully to try to find the optimal solution to each level, Uplink revolves around fast movements to get the job done. You get in, you screw around with stuff, you erase your tracks, and you get out. If you know what you're doing, a mission could be as short as two or three minutes, compared to HE's half-hour time commitment. This speedy in-and-out sequence creates an atmosphere that is, quite frankly, absolutely terrifying. You know you've got to do so much in so little time, and not being fully prepared (or even knowing if you're fully prepared) makes every new mission really exciting.
Note that I said "every new mission" though. One definite downside to this game (at least, in the three hours or so that I've played of it so far) is that especially in the early stages, you'll find yourself grinding a lot. To raise enough funds to get better equipment and to increase your rating level in the Uplink community, you'll find yourself repeating a lot of similar missions over and over again. Granted, they're quick, and they really reinforce the concepts you're learning, but you start to get a bit tired of these quickly.
As I'm writing this, I'm realizing that I should have written this entire post backward. Did I mention Uplink is much easier to get a hold of than HE? You'll recall that when I wrote about HE, I noted that it tended to throw all the concepts for the entire game upfront in a tutorial level, which isn't easily revisited later on. Uplink avoids that problem with these short, simple missions that let you get familiar with your tiny toolbox. New concepts are slowly worked into the game through very subtle trial-and-error. For example, one mission I just started requires you to alter some information in a database, which I've done before. However, entering into the level, you have no idea that the information is guarded by a proxy (whatever that is). You're able to escape without any consequence, then get a proxy-bypassing tool and try again. It's not exactly "teaching" you the concepts, but you "learn" them through roundabout methods.
I can only think of one other major remark to make about Uplink, and it regards the game's interface. Almost all of the game is controlled by the mouse, except for when you need to edit text directly (file tampering, etc.). This is way more convenient than trying to remember HE's cavalcade of DOS commands, but still a bit of a hassle in certain places. To input text into a box, you can't just click on it, you have to keep your mouse hovered over the box. Moving the mouse away "deselects" the box and you have to move the mouse back to regain focus. In an interface that's otherwise fairly workable (well, it's also a pain having eight different information sources pop into the same spot, meaning you can't have two open at once), it seems weird that such an annoying problem would be considered passable.
If you're looking for a final verdict between HE and Uplink, I can only say that my decision is split, but I'd lean more toward Uplink. Uplink is a way more easily-digestible game than HE, giving you smaller tasks to accomplish (which you can leave in the middle of, by the way). On the other hand, it moves a bit too quickly to be really involved in the game beyond quick commands, so it's missing a certain puzzle element HE has, but that quickfire presentation gives you a more thrilling experience. I'd say it's worth a go, if it goes up on sale again (we'll see what sales the upcoming summer holidays will bring).