What do you like best?
I've been an early user of AIB and I keep checking on other products and email clients just to see if anything else matches up. Short answer: nothing has.
I've lost count, but I think I've used it for close to a decade and AIB has grown over the years to a slick feature-packed add-on to email in a browser. Its best features: ability to treat emails as to-dos and follow up on them as you need, notes and mini to-do lists that can be tacked on to individual emails, superb tagging that enables quick retrieval, awesome integration with GMail in your browser and quick keyboard shortcuts.
What do you dislike?
Nothing really. It's a browser extension which rules out using it in an email client, which might be a deal breaker for some, but I haven't yet met an email client (and believe me, I've tried a lot!) that works well enough to woo me away from AIB.
Recommendations to others considering the product
If you're looking for a browser extension to manage your email and you're happy with living with email in your browser then AIB is outstanding, combining a bunch of features that you'd otherwise need multiple extensions for.
What business problems are you solving with the product? What benefits have you realized?
I run a small business with just me and my wife and the nature of our business requires waiting a lot on email responses from others - clients, service providers etc. (like most businesses do). AIB helps me track which emails have been pending for responses, which are due, overdue, which need to be nudged, which need to be kicked. I can set due dates and track which conversations are on track and which need attention. All very easily, simply and without great overhead in terms of time and effort. It has its roots in the GTD philosophy and although I don't actively use GTD, its principles are useful and common sense based. If Inbox Zero is your thing (it is mine), then AIB is superb at triaging your inbox and slotting things away neatly as to-dos or "dones".