
die Empfindung
sensation
A physical perception conveyed through the senses, such as cold, warmth, pain, or touch.
Nach der Betäubung hatte er keine Empfindung mehr im Bein.
After the anesthesia, he no longer had any sensation in his leg.
sensation / feeling
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sensation
A physical perception conveyed through the senses, such as cold, warmth, pain, or touch.
Nach der Betäubung hatte er keine Empfindung mehr im Bein.
After the anesthesia, he no longer had any sensation in his leg.

feeling
An inner feeling, an emotion, or a hunch. It describes an emotional state or an intuitive perception.
Ich hatte die seltsame Empfindung, dass etwas nicht stimmte.
I had the strange feeling that something was wrong.
'Die Empfindung' (sensation/feeling) is the noun corresponding to the verb 'empfinden' (to feel or perceive). If you know that 'empfinden' means 'fühlen' (to feel) or 'wahrnehmen' (to perceive), you can understand the meaning of the noun more easily.
Er konnte seine Empfindungen nur schwer in Worte fassen.
He found it difficult to put his feelings into words.
You can recognize the meaning from the context: If it is about the body, senses, or medicine, a physical sensation is meant. If it is about intuition, feelings, or a hunch, an inner feeling is meant.
Er spürte eine schmerzhafte Empfindung im Rücken.
He felt a painful sensation in his back.
For physical perception, 'Empfindung' (sensation) is often combined with adjectives such as 'kalt' (cold), 'warm' (warm), 'taub' (numb), or 'brennend' (burning). The phrase 'eine Empfindung von Kälte/Wärme' (a sensation of cold/warmth) is also common.
Nach dem langen Spaziergang im Schnee hatte sie eine taube Empfindung in den Zehen.
After the long walk in the snow, she had a numb sensation in her toes.
To express an inner feeling or a hunch, the fixed structure 'die Empfindung haben, dass ...' (to have the feeling that ...) is often used. It is followed by a subordinate clause.
Ich habe die Empfindung, dass wir hier nicht willkommen sind.
I have the feeling that we are not welcome here.
feeling / sense
Here, das Gefühl generally refers to the ability to perceive something through the skin. Die Empfindung, by contrast, is a specific physical perception, such as warmth, cold, or pain. Use Gefühl for the ability to perceive, and Empfindung for a particular physical stimulus.

Nach der Behandlung spürt sie eine warme Empfindung in ihrer Hand.
After the treatment, she feels a warm sensation in her hand.
Nach der Verletzung kehrt das Gefühl in ihrer Hand langsam zurück.
After the injury, feeling is slowly returning to her hand.