Silence, quiet, rest, and tranquility

There are two German nouns that are translated as "silence"—die Stille and das Schweigen.

The main difference is that das Schweigen implies a conscious choice by a person to remain silent, whereas die Stille can occur naturally without human intervention. Let's have a look:

Vom Trubel des Narzissenfestes geht es jetzt nach Osttirol in die Stille und Einsamkeit der Bergwelt.

From the hustle and bustle of the Daffodil Festival we are now heading to East Tyrol into the silence and solitude of the mountain world.

Captions 1-2, Die letzten Paradiese: Die Schönheit der Alpen 1
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There is the related adjective still, which means "silent":

Stille Nacht, heilige Nacht

Silent night, holy night

Caption 2, Weihnachtslieder: Stille Nacht mit Gabriela, Mira und Ralph
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It's important to note that while das Schweigen is related to the commonly used verb schweigen ("to remain silent"), the verb stillen ("to breastfeed") does not relate in the same way to the noun die Stille.

Sie haben die Pflicht zu schweigen.

You have a duty to remain silent.

Caption 26, extra 3: Sprachpolizei gegen falsches Deutsch
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Die Nacht so laut, doch in mir schweigt's.

The night so loud, but inside me it's silent.

Caption 11, Joris x Lotte: Wer wir geworden sind
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The noun die Ruhe, its related adjective ruhig, and the verbs ruhen and ausruhen refer also to a quiet, calm, restful state, though not necessarily completely silent.

Hinter dem Schloss finde ich Orte der Ruhe und Erhabenheit.

Behind the castle I find places of tranquility and [the] sublime.

Caption 18, Architektur: Karlsruher Brunnen
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Dann verordne ich mir immer ganz viel Ruhe.

Then I always prescribe myself lots of rest.

Caption 25, Deutsch mit Eylin: Erkältungszeit
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Sei ruhig!

Be quiet!

Caption 21, Deutsch mit Eylin: Der Imperativ
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Zwei Männer ruhen sich aus, einer überwacht das Gelände

Two men will rest, one will keep guard of the site,

Caption 56, Es war einmal... der Weltraum: Die Saurier
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Finally, another adjective that also can mean "quiet" is leise. It refers to a low volume.

Oh, der ist aber leise, man hört ja gar nichts.

Oh, it's so quiet, you can't hear anything.

Caption 8, heute-show: Werden Elektroautos das Klima retten?
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Further Learning
You can find all of these words on Yabla German and watch the Piggeldy und Frederick episode on die Stille for a humorous take on the topic.