schmerzen
to ache
To cause or feel physical pain.
Nach dem langen Lauf schmerzt mein Knie.
After the long run, my knee aches.

to hurt / ache
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to ache
To cause or feel physical pain.
Nach dem langen Lauf schmerzt mein Knie.
After the long run, my knee aches.

The verb 'schmerzen' (to hurt / ache) is derived directly from the noun 'der Schmerz' (pain). When something 'schmerzt' (hurts/aches), it causes pain. Both words are used for physical and emotional suffering.
Der Verlust seines Freundes bereitete ihm großen Schmerz.
The loss of his friend caused him great pain.
The case helps identify the type of pain. With physical pain, the person often appears in the dative ('Mir schmerzt der Rücken' — my back hurts). With emotional pain, the person appears in the accusative ('Sein Verrat schmerzt mich' — his betrayal hurts me).
Ihr schmerzte der Kopf von dem Lärm.
Her head ached from the noise.
The verb 'schmerzen' (to hurt / ache) can also be used impersonally with 'es' as the subject. This is typical in questions at the doctor's office when asking about the location of the pain.
Können Sie mir zeigen, wo genau es schmerzt?
Can you show me exactly where it hurts?
to do
'weh tun' and 'schmerzen' can both express that someone has physical or emotional pain. 'weh tun' is very common in everyday language and sounds direct and simple: 'Der Kopf tut weh' (the head hurts), 'ein Satz tut weh' (a sentence hurts). 'schmerzen' sounds somewhat more formal or serious and is often used with body parts or strong feelings. Learners may confuse the words because their meanings are very similar. In normal conversation, people usually say 'Mein Knie tut weh' (My knee hurts). In a report or for a stronger expression, 'Mein Knie schmerzt' (My knee aches/hurts) can sound more natural.

Beim Gehen schmerzte ihr Fuß noch immer.
Her foot still hurt when she walked.
Nach dem Training tut Nico der Rücken weh.
After training, Nico's back hurts.