der Schmerz
pain
An unpleasant physical sensation caused by an injury or illness.
Nach der Operation hatte er starke Schmerzen im Bein.
After the operation, he had severe pain in his leg.

pain
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pain
An unpleasant physical sensation caused by an injury or illness.
Nach der Operation hatte er starke Schmerzen im Bein.
After the operation, he had severe pain in his leg.

Physical pain is often used in the plural in German ('Schmerzen'), while emotional suffering is almost always in the singular ('der Schmerz'). This usually makes the meaning easy to recognize.
Sie hat seit gestern starke Schmerzen im Rücken.
She has had severe back pain since yesterday.
For physical complaints, the plural form 'die Schmerzen' (pain) is used very frequently, even when only one part of the body is affected. The question 'Wo haben Sie Schmerzen?' (Where do you have pain?) is typical at the doctor's.
Ich habe seit Tagen Schmerzen im Knie.
I have had pain in my knee for days.
'Schmerz' (pain) is often combined with a body part to form a new word that specifies the type of pain. Examples are 'Kopfschmerzen' (headache), 'Bauchschmerzen' (stomachache), or 'Rückenschmerzen' (back pain).
Nach der langen Autofahrt hatte er starke Rückenschmerzen.
After the long car ride, he had severe back pain.
Emotional pain is often combined with verbs such as 'empfinden' (to feel/experience) or 'fühlen' (to feel). The cause of the pain is often expressed with the preposition 'über' (+ accusative).
Er fühlte einen tiefen Schmerz über den Verlust seines Freundes.
He felt deep pain over the loss of his friend.
suffering
'das Leid' and 'der Schmerz' both describe something very unpleasant. 'Schmerz' is used very often for the body, for example with an injury, but also for emotional grief. 'Leid' is usually bigger and more general: it means deep sorrow, misfortune, or prolonged suffering. If someone has a headache, you normally say 'Schmerz'. If you are talking about a person's difficult life or grief, 'Leid' often fits better.

Der Schmerz im Knie wurde beim Treppensteigen stärker.
The pain in his knee got worse when he climbed the stairs.
Nach dem Unfall war das Leid der Familie groß.
After the accident, the family's suffering was great.