annoyance / trouble
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annoyance
A feeling of dissatisfaction, anger, or irritation about something that is not going as desired or is perceived as disruptive.
Sein Ärger über die ständige laute Musik des Nachbarn wuchs von Tag zu Tag.
His annoyance about the neighbor's constant loud music grew from day to day.
'Der Ärger' (annoyance/trouble) belongs to a word family with the reflexive verb 'sich ärgern' (to be annoyed) and the adjective 'ärgerlich' (annoying/irritating). All three describe a feeling of anger or dissatisfaction.
Ich ärgere mich über das schlechte Wetter.
I'm annoyed about the bad weather.
The prepositions are important: you have 'Ärger mit' (trouble with) a person or thing (dative). You feel 'Ärger über' (annoyance about) a situation or action (accusative).
Sie hat oft Ärger mit ihrem alten Auto.
She often has trouble with her old car.
In the colloquial meaning 'Probleme' (problems), 'Ärger' (trouble) is often used with the verbs 'haben' (to have), 'bekommen' (to get), or 'machen' (to make/cause).
Wenn du zu spät kommst, bekommst du Ärger mit dem Chef.
If you come too late, you'll get in trouble with the boss.
The meaning 'Schwierigkeiten' (difficulties) or 'Probleme' (problems) is typical of colloquial language. In formal or written contexts, you would more likely use 'Schwierigkeiten' (difficulties) or 'Unannehmlichkeiten' (inconveniences).
Ich will keinen Ärger mit der Polizei.
I don't want any trouble with the police.
problem
'das Problem' and 'der Ärger' can both have to do with a difficult or unpleasant situation. 'das Problem' emphasizes that something is not working, is unclear, or needs a solution. It can be factual or personal. 'der Ärger' often means the feeling that you are angry or dissatisfied. Colloquially, 'Ärger' can also mean trouble or difficulties. That is why learners may confuse the words. If you are talking about the thing that needs to be solved, 'Problem' is usually more precise. If you mean the unpleasant reaction or the stress, 'Ärger' fits better.
Wegen des falsch geparkten Autos gab es in der Straße viel Ärger.
Because of the incorrectly parked car, there was a lot of trouble on the street.
Beim Ausfüllen des Formulars hat Herr Becker ein Problem mit der Adresse.
While filling out the form, Mr. Becker has a problem with the address.