What's the difference and who do I need to get permission from to use each?
A Sample is a piece of recording reused in another recording. When you sample something, you are literally taking a snippet of a original recording and adding that snippet into a new recording.
An interpolation on the other hand, refers to when a piece of a music is replayed in a new recording and meant to sound exactly the same as the original recording. Rather than literally copy and pasting a snippet from a recording, you are re-playing and re-recording the underlying composition into a new recording.
For example, if you replay the melody from "Lollipop" by Lil Wayne into your own song, this is considered an interpolation. However, if you literally copy and paste the melody from the original recording into your new song, this is considered sampling.
It is important to understand the difference between the two, as the clearing process for interpolations and samples are slightly different.
When you sample something, you are required to get permission from both the owner of the Master Recording (normally a record label) for a Master Use License and also the owner of the underlying composition (normally a publisher) for a copyright license. This is because you are featuring both the recording and the underlying composition in the new musical work. However, if you are just doing an interpolation of a song, you only need to get permission from the owner of the underlying composition since you are not featuring the original recording in your new song, and just featuring the underlying composition.