Well, I'll start by saying that (at least to my knowledge, in the USA) there are no certificated/PMA'd/TSO'd LiFePO4 batteries currently available on the market for use in certificated airframes.
That means if you're using a type-certificated aircraft (or an LSA that didn't ship with one originally and for which there is no manufacturer-approved procedure to install one) you're basically out of luck unless you can convince your local FSDO to approve it as a one-off modification to your airframe.
That said, there are some people using them in homebuilts. This guy, for example has used one on his SQ2000 to replace the starter/cranking battery, and there's a company called AeroLithium that sells aviation lithium batteries (albeit uncertified) which some folks seem to be using in (experimental) Carbon Cubs with a good deal of success.
As to the batteries themselves, my aircraft is certificated, I've got no desire to take it experimental, and my relationship with the local FSDO is not one where I could just walk in and say "I want to stick one of these in my plane. Sign off on the 337 form KTHXBYE!" so this is all just theoretical:
Operationally, LiFePO4 is a relatively safe chemistry.
Unlike conventional lead-acid batteries it's not going to spit sulfuric acid on your plane during charging or hard maneuvering. Unlike other Lithium-based chemistries (say Lithium-Cobalt of Dreamliner fire fame) it's not as prone to blowing up or catching fire if short-circuited or otherwise abused (though puncturing it is still a Bad Thing).
Note however that as LiFePO4 is a lithium based chemistry water may not be the best extinguishing agent for a ruptured/burning battery. One of the potential decomposition products of a burning or ruptured LiFePO4 battery in the presence of water is Hydrogen Fluoride (HF), which is rather nasty stuff.
Handling and Charging is generally normal - as in you can connect it to your aircraft charging system with appropriately-rated breakers/fuses.
Checking cell balance is recommended periodically, but without adequate information on the use of these batteries in aircraft we can't say what "periodically" means -- Every 10 hours? Every 50? Annually? (If installing one of these in a homebuilt I'd start with every 10 hours and work from there).
Robustness should be superior to lead-acid batteries in all respects (mechanical and electrical).
For use in an aircraft a sturdy housing around the cells would be a Must Have to prevent the battery from being ruptured in the event of an incident, but that could be produced reasonably.
Electrically LiFePO4 has a lower self-discharge rate, better tolerance for overcharging, and better (flatter) discharge curves - all Good Things. It is also stable over a wide range of temperatures, and at least theoretically will outlast lead-acid batteries on a charge/discharge cycle and calendar-year basis (though again, without real-world utilization data we can't really say anything for sure here).
Cost-Benefit is hard to qualify.
LiFePO4 batteries can be had cheaper than "aircraft batteries", largely owing to the fact that they're not approved for use in certificated aircraft.
Because of their higher energy density they're also substantially lighter for the same amp-hour capacity, which is always attractive when we're talking about aircraft applications.
All of those benefits aren't worth a warm bucket of spit to me because I can't install one in my airplane (though I'd certainly consider it seriously if it were a viable option). If you can install one in your airplane as a replacement for your main battery it may well be worth considering.