
hartnäckig
persistent
Describes something that is difficult to remove, heal, or solve, such as a stain, an illness, or a rumor.
Dieser Rotweinfleck auf dem Teppich ist wirklich hartnäckig.
This red wine stain on the carpet is really persistent.
persistent / stubborn
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persistent
Describes something that is difficult to remove, heal, or solve, such as a stain, an illness, or a rumor.
Dieser Rotweinfleck auf dem Teppich ist wirklich hartnäckig.
This red wine stain on the carpet is really persistent.

persistent / stubborn
Describes a person or their behavior as persevering, persistent, and unyielding in pursuing a goal. This can be meant both positively (determined) and negatively (stubborn).
Trotz mehrerer Absagen blieb sie hartnäckig und bewarb sich erneut.
Despite several rejections, she remained persistent and applied again.
The word is a compound of 'hart' (hard) and 'Nacken' (neck) with the adjective ending '-ig'. The idea of a 'harter Nacken' (stiff neck) that cannot be bent figuratively describes the quality of being stubborn, unyielding, or resistant.
Er blieb hartnäckig bei seiner Meinung, obwohl alle dagegen waren.
He stubbornly stuck to his opinion, even though everyone was against it.
In this meaning, 'hartnäckig' (persistent) is often used to describe things that you want to get rid of but that remain. Typical examples are 'ein hartnäckiger Fleck' (a stubborn stain), 'ein hartnäckiger Husten' (a persistent cough), 'hartnäckiger Schmutz' (stubborn dirt), or 'ein hartnäckiges Gerücht' (a persistent rumor).
Ich kämpfe seit Wochen gegen eine hartnäckige Erkältung.
I have been fighting a persistent cold for weeks.
When 'hartnäckig' (persistent/stubborn) describes a person, the meaning can be positive (determined, persevering) or negative (stubborn, unreasonable), depending on the context. The situation and the wording in the rest of the sentence show which evaluation is meant.
Die Journalistin stellte so lange hartnäckig Fragen, bis sie eine Antwort bekam.
The journalist kept asking persistent questions until she got an answer.
chronic
Both words can describe a long-standing illness or ailment. Chronisch emphasizes that an illness lasts a long time or keeps coming back, while hartnäckig highlights that it is difficult to get rid of or cure. Depending on the intended nuance, either word may be appropriate.

Die Hautkrankheit ist hartnäckig und lässt sich trotz Behandlung nur schwer heilen.
The skin disease is persistent and is difficult to cure despite treatment.
Die Hautkrankheit ist chronisch und kehrt immer wieder zurück.
The skin disease is chronic and keeps coming back.