Well, lets compare the two devices and how you would use them.
Fundamentally, they will both do the same thing in that they will allow you to plug two microphones into it, convert that mic's signal, and send it to the computer via the USB connection. Software on the computer will see the two physical connections and allow you to record them separately. It provides a headphone port for you to monitor the sounds, so you aren't bothered by latency issues.
Either device will do that, so the question is "What do I get from one vs the other?"
With the Art Dual, you do all of your recording setup in some sort of DAW software (Audacity, Garage Band, Reaper, Pro Tools, Cube Base, etc...) The software is just a computer program that acts like a physical mixing board. Plug inputs into tracks, set your record levels, pan them left or right, mute or solo tracks, add effects to them, etc, just like you would if you were plugged into a physical mixing board. You have to set all of that with the mouse and keyboard, though, so it's not as convenient as having a physical control surface in front of you.
With the Alesis, you have some of those controls along with the interface. It offers a two band EQ, pan controls, level controls, and a high pass filter on each channel. Nice to have stuff.
So if we were to consider trade-offs (I have no experience with either device, so these are purely hypothetical) and pros and cons...
Potential ART Pros: Smaller and more portable. Higher quality pre-amps and digital converters because that is all that it does. Completely USB powered (more portability).
Potential ART Cons: Only two channel. Dependent on DAW software for everything (your computer probably isn't as portable as this thing is).
Potential Alesis Pros: On board EQ and controls. Four channels. Usable as a live mixer (separate main and monitor outs, and also a really handy feature).
Potential Alesis Cons: Lower quality components. Requires external power (less portable).
So I guess the easiest way to understand it is that we are really talking about two separate kinds of devices here. The "interface" is what converts microphone signals to something the computer can understand, and allows the computer to understand that it might be receiving more than one signal at a time. The "mixer" is what blends audio together into the final stereo output that you hear in the speakers. The ART is only an interface, and you have to provide your mixer (usually in the form of a DAW) while the Alesis provides an interface, along with some basic mixing functionality.
Did that help, or did it muddy it up even more?
I went and read a lot of reviews on a lot of sites for these two specific devices, and just as an exercise, I think the tradeoff notions are valid. Lots of people on lots of sites complain about noise problems with the Alesis but praise it for it's flexibility and utility in live mixing situations. The ART gets tons of praise for it's quality pre-amps and ease of use.
Someday we'll win this thing...