Both of the words above mean "both," but it's important to use them in the correct contexts and with the correct grammar. 

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The word beide is plural and often refers to people. It can be used as a pronoun...

Beide sind jetzt vier Stunden und sieben Minuten unterwegs.

Both have now been traveling for four hours and seven minutes.

Caption 31, Galileo Zug vs. Flugzeug: Von München nach Berlin

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Beide haben sich entsprechend dazu geäußert.

Both have commented on it.

Caption 49, Archäologie: Der Fund der Frankfurter Silberinschrift

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or as a determiner:

Gut, wenn das jetzt beide Spieler wirklich genauso machen, dann kommt so ein furchtbar langweiliges, symmetrisches Brett dabei raus.

OK, if both players really do the same thing, then you end up with a terribly boring, symmetrical board.

Captions 18-19, Schach: Regeln, Strategie und Taktik

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The phrase wir beide can be translated as "we both," but also "the two of us."

Und ja, wir beide werden uns nie wieder sehen.

And yes, the two of us will never see each other again.

Caption 19, Cro: Bye Bye

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Here are a few more examples with beide as a determiner in sentences that don't refer to people:

Und das gilt für beide Seiten gleichermaßen.

And that applies to both sides equally.

Caption 25, Brexit-Votum: Merkel warnt vor Spaltung Europas

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Beide Sätze sind im Perfekt.

Both sentences are in the perfect tense.

Caption 22, Deutsch mit Eylin: Das Perfekt

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Although beides refers to two things, it is grammatically singular and neuter, and takes the third person singular form. And it should not be used to refer to people, nor as a determiner. In the sentences below, beides refers to two words, two tasks, and two continents respectively.

Weißt du, „Möhre“, „Karotte“, das ist beides richtig. -Mhm.

You know, "carrot," "carrot," both are correct. -Mm-hm.

Caption 23, Mein Weg nach Deutschland: Bei der Arbeit

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Mit Emotionen oder technischen Problemen umgehen: Man muss beides können.

Dealing with emotions or technical problems: You have to be able to do both.

Captions 8-9, Galileo: So hart ist der Job im Callcenter

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Und es ist beides total unterschiedlich, aber super...

And both are totally different but great...

Caption 44, Anna: Am Strand in Mexiko

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Further Learning
You can find more examples with beide and beides on Yabla German.