Same shape, different rating scheme
The size code describes geometry, and on geometry the two are near enough identical. What differs is the rating scheme behind them: how much load the tire is certified for, at what pressure, and under what assumptions about duty.
An LT tire is built and rated for light truck service. It runs at higher inflation pressures, carries a load range letter, and often prints two load indices for single and dual fitment. A P-metric tire is rated for passenger car service.
The derating
When a P-metric or Euro-metric tire is fitted to a light truck, SUV, pickup, van or minivan, its rated load is divided by 1.10 before it counts. Both tire manufacturer technical bulletins consulted for this site attribute the factor to federal motor vehicle safety standards, and both apply it to Euro-metric as well as P-metric.
An LT tire takes no such reduction, because light truck duty is what it was rated for in the first place. That single asymmetry is why a P-metric size cannot simply replace the LT size a vehicle specifies, even though the numbers look identical on the sidewall.
What it means in practice
If the vehicle placard specifies an LT tire, replace it with an LT tire. Going to P-metric in the same size reduces the rated load capacity regardless of what the load index says, and it is exactly the situation the derating exists to describe.
Going the other way, from P-metric to LT, adds capacity and adds weight, stiffness and rolling resistance with it. It is a real choice for a vehicle that tows or hauls near its rating, and an unnecessary one for a vehicle that does not.