Well, disclaimer, I'm not an expert at all, but this is what I've managed to gather over the past six months.
NMC batteries are more energy-dense, and so have a longer range per kg/litre of battery weight/size. They also charge faster. This makes them more suitable for longer-range cars and cars that will be required to charge quickly (which tend to be the same thing). However this brings downsides. They aren't so long-lived (though that's relative, they're still pretty long-lived) and they're more finicky to look after. It's recommended not to leave them sitting around at either very high or very low states of charge for very long (under 20% or over 80%), so normally one would operate between these states unless preparing for and undertaking a long journey, when the battery won't be sitting around at any particular rate of charge. It is however recommended to charge them to 100% on an AC charger and let the cells balance by leaving the car plugged into the charger for as long as it takes, at least once a month. They also contain cobalt, which is known as a "conflict mineral" and is not beloved of environmentalists or human rights campaigners.
LFP batteries are less energy-dense, so have a shorter range for the same size/weight, and charge at only about 2/3rds the speed. However they are very long-lived and easier to look after. They don't mind being charged to 100% and left there, as the over-potential that causes electrolyte degradation in the NMC batteries is not present. You can just whack them up to 100% any time you want to. They like to be balanced at least once a week, but since you'll always be charging to 100% this happens without any extra effort. (They will balance every time they're taken to 100% on an AC charger.) They don't have any cobalt at all, so LFP owners can feel superior in this respect to anything else on the road.
If your normal daily mileage doesn't exceed the capacity of your battery and you can charge at home, the LFP is a bit of a no-brainer. You can practically charge it as you would your phone, not thinking about it too much. You can always leave home with enough charge to get you back, rinse and repeat. This applies so long as you're relaxed about your occasional long journeys needing an extra stop (because of the lower capacity) and the stops being a bit longer (because of the slower charging). This is why I went for the LFP.
On the other hand if you need to do journeys beyond the range of the car where time is pressing, go for the NMC. Also, if you can't charge at home and don't have easy access to an AC charger within walking distance, I think there's a lot to be said for the NMC. You can operate within the 20-80% range on rapid chargers (or grazing on AC chargers without going to 100%) most of the time, and you won't have to wait around too much. You only have to organise to leave the car to balance on an AC charger once a month. It will be easier to fuel more in the manner of an ICE car.
No doubt someone will be along to explain that this is wrong, or at best overly simplistic, but it's my current understanding.