What are the advantages and disadvantages between the two or the features that comes with price difference? Is it really worth paying $350 on average more for the T4i?
Is it worth is a difficult question to answer as it depends greatly on what you would be using it for.
There are a few primary areas of differences to review:
The AF system has been upgraded to a 9-point cross-type. This means that you can accurately focus using any of the 9 AF points and they will work together to determine a solid focus. On the T3i, you only had a cross-type center point focus. This is going to serve a big advantage in low-light focusing. There is also the new "Hybrid" focus system that allows phase AF even in Live View. I wish I had this in my 7D because it significantly increases AF speed and prevents focus hunting.
The ISO on the high end has been doubled. While the sensor is relatively the same, the new Digic processor is going to be able to reduce noise better than the T3i. Don't expect a significant improvement though (or usable results at the high end).
The new AF hybrid system is a huge advantage in the movie mode. They've also added stereo recording which isn't incredibly impressive.
Significant upgrade for many users in my opinion. The jump to 6FPS closes the gap with the XXD series, which used to be a large deciding factor of moving up. This will make the difference in grabbing the decisive shot in a kids sports game. I'd consider it for this alone.
One of the biggest reasons we use a DSLR is for the lenses, but now we have the mirror-less that are filling in that gap. Our new excuse for using the DSLRs is the buttons for quick access. Touch screen defeats that purpose. You can navigate fine, if not faster without a touch screen. It's not an improvement, simply a "feature".
The value of $350 depends on you. You are right that is steep for the T3i vs T4i.
The main difference is that the T4i features a revised and actually unique sensor. For video, the new sensor has built-in phase-detect autofocus which speeds up focusing during video capture. This is obviously only advantageous if you shoot video and you do so with autofocus.
For photos, there is slightly lower image noise and faster burst rate of 5 FPS vs 3.7 FPS. The camera normally uses phase-detection for capturing images and it can do this too in Live-View as well with the new sensor. Again, how much this is worth it depends on your use of this feature.