In what ways does the Arkham Horror Toolkit app improve or detract from the experience?
Since I've never used the app before (I have android devices, not Apple) I can only answer from what I've seen with other apps like this and tailor it to what the Toolkit claims to do.

Since it has a full list of every event card and gate card, it does save on the setup of the game significantly.  However, since it doesn't have the Mythos deck set up and tracked, it doesn't help the app very much in that regard.  You still have to pull out cards and, when you're setting up, you might as well pull everything out and set it down instead of leaving just the event cards and gate cards.

It also gives you a dice roller, which I don't see as really a time-saver.  Plus, it feels better to roll dice physically.

The investigator tracking sheet is nice to remember what your current stats are for all of your skills, how many clues you have, how much damage you've taken, ect.  It really clears up the clutter of the board.  Down side?  It doesn't show you everything about your character, so you still need the card out.  And it only tracks your character, not all the others.  You'd need to buy the thing for a lot of people.

Lastly, it gives you maps of each of the boards (so long as you buy them as add-ins).  This helps in reading the board for yourself, but you have to have the board out for you in the first place, so I don't see this as a time saver either.