What is the difference between power inductors and decoupling inductors?
Those two data sheets make it difficult to compare like with like, as the power inductors have two rated currents (inductance and thermal), whereas the decoupling inductors have a single rated current, and a minimum Q.

In my designs, I have two different applications, both of which require things called inductors. They both require a certain current to be supported while maintaining a minimum inductance. 

The first application is in a power supply filter network, where some degree of dissipation in the ferrite at high frequency is positively beneficial. It damps down resonances between components, and absorbs RF signals that I don't want crawling between one module and another. Even those inductors look a bit 'lively' for my tastes, and I usually end up using 'impeders', which have a specified loss at RF.

The second application is energy storage, in dc-dc converters for instance. Here, any power dissipation in the ferrite is a bad thing, it all contributes to overall inefficiency in a converter design.

You can't really optimise designs for either of these applications from a brief datasheet alone. You need a detailed data sheet, and then investigation on a bench with the appropriate test gear. Though using components from the manufacturer's suggested range is a good start.