Option 1. Access EssayTagger from two different browsers (e.g. Chrome and Firefox). This will allow you to launch the grading app in each browser, one for each revision. You can take advantage of computers' built-in shortcut to quickly jump back and forth between open programs (ALT-TAB on PC, CMD-TAB on Mac). If you're on a more recent Mac you can fullscreen each browser and use the four-finger swipe left and right to swap between the workspace panes. This is how I do all of my multi-screen work.
Option 2. Open the student's "public link" to view the graded output of their earlier draft. Then launch the grading app to evaluate the new draft.
As for a built-in draft comparison feature... that's too difficult. Imagine that a student just moved a paragraph to fix an organizational issue but the content didn't change. That's not too bad for a program to figure out which pieces are related to each other. But now let's say she moves a piece of evidence out of that moved paragraph and inserts it elsewhere. Now things are getting messy. And then she rewrites one portion of the sentence that introduces that evidence. Now we have total chaos--the program won't be able to figure out where that the original comment related to that evidence should go.
If you ever use Word's track changes feature, you'll see that it quickly becomes useless when document elements move around. Basically the whole doc ends up highlighted as changed, even though all you did was shuffle things about a bit.
It's just one of those problems that ends up being very difficult for software to deal with.