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If you'd like for Spotify to take up your name from Facebook as a display name, first make sure that they are properly connected. This page explains how to link your two profiles. Note that this step is only necessary if you haven't created your Spotify account through Facebook.
After the connection is established, it's a good idea to make sure that the option to process your Facebook data is enabled. You can learn how to do this here. Keep in mind that it can take up to 48 hours for you display name to get updated.
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“Music is a language that doesn’t speak in particular words. It speaks in emotions, and if it’s in the bones, it’s in the bones.” - Keith Richards