There are 3 main advantages to leather. The first being its natural water resistant properties. You can be wearing your boots for long periods of time and if your feet become wet not only can it be uncomfortable but it can also lead to damaging medical issues.
The second benefit is strength. Leather is a very tough material and can withstand a lot of contact with abrasive terrain. This means your boots or shoes will last longer and are less likely to get damaged or rip.
The third advantage is quality. Leather is usually used in the higher range of boots, ensuring that you receive a high level of quality throughout.
This does however have a disadvantage, which is cost. Leather walking boots and shoes do tend to be more expensive than other materials.
The other issue you may find with leather is that it will require some time to break in. The leather is initially quite stiff and can rub your feet if used for long periods of time. The way to prevent this is to gradually wear your boots for increasing amounts of time to slowly mold the leather to the shape of your feet and making it more supple.