Frame rate and Refresh rate are sometimes used interchangeably and while they are very related, they do not mean the same thing.
Frame rates simply mean the amount of frames (still pictures) that are pushed to the display by the GPU. It is measured in frames per second (fps).
Refresh rates on the other hand are how quickly a display can update itself to show new images on the screen. It is measured in Hertz (Hz).
Similarity
Frame rate and Refresh rate both refer to the number of times that still images are
shown on a display in one second.
Read: Frame rates (fps) explained
Difference
The frame rate is handled by the GPU while the Refresh rate is handled entirely by the display. They do not control each other. The frame rate also greatly depends on the media (game or movie) being played. If you have a 60Hz display and your phone can only give out 30fps, all you effectively get is 30fps. This is because your display is going to be redrawing the same image over and over until the GPU sends a new one.
Read: What are Refresh rates (Hz)?
How do they work?
When playing a game on your phone, for example, the GPU will load the data from your storage and render images frame by frame. These images are then pushed to your display. The display then accepts the information from the GPU and shows it for you to see.
Ideally, the frame rate and refresh rate should be equal (1 fps is equal to 1 Hz). This means that as each frame is pushed from the GPU, the display refreshes to update it on the screen. In the real world though, things are not so ideal. The base refresh rate for most displays is 60Hz.
Interpolation
Now when the smartphone (game or movie) pushes 30 frames in one second (30fps) to a 60Hz display. This means that the display is two times faster than the game or movie. When this happens, the display would then fill in the space with fake frames generated from existing frames. This is called interpolation.
The display would interpolate to fill in the missing frames. This is what happens when movies (24fps) and most games (30fps) are being played on your screen.
Video credit: Wikipedia
In the video above, the frame on the left is not interpolated while the one on the right is interpolated. Interpolation helps to make motion pictures more fluid and easy on the eyes. It also removes motion blur as well as jumpy images.
Frame rate vs Refresh rate
Triple-A game titles are some of the only media that can push 60 frames to a display in one second (60fps). This is ideal for 60Hz displays because you are getting a new image from the GPU every time the display refreshes. In this case, 1fps = 1Hz. This is also the case when 90fps is pushed to a 90Hz display.
If you have a game of 90fps and your display can only refresh at 60Hz. What you are getting is 60fps. This is because your display can only update itself 60 times in one second. When this happens, you are going to missing those remaining 30 frames as they are not going to be displaying on your screen.
Screen tearing
In those non-ideal cases where 1fps is not equal to 1Hz, the GPU and the display may not sync properly. This usually results in screen tearing. Screen tearing is when two images from different frames appear on the screen at the same time.
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