
fällig
due
Describes a debt or bill that must be paid at a certain time.
Die Miete ist immer am Ersten des Monats fällig.
The rent is always due on the first of the month.
due
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due
Describes a debt or bill that must be paid at a certain time.
Die Miete ist immer am Ersten des Monats fällig.
The rent is always due on the first of the month.

due
Describes something that is expected or must happen at a certain time.
Die nächste Inspektion für das Auto ist in zwei Monaten fällig.
The next inspection for the car is due in two months.

to be due
Colloquial for something that is necessary or deserved as a consequence, often in a negative sense.
Nach dieser frechen Antwort war eine ordentliche Standpauke fällig.
After that cheeky answer, a proper telling-off was due.
'Fällig' (due) is almost always used with the verb 'sein' (to be) or 'werden' (to become). The structure is typically: [something] ist/wird [when] fällig ([something] is/becomes due [when]).
Die nächste Rate für das Auto wird am Monatsende fällig.
The next installment payment for the car is due at the end of the month.
In financial contexts, 'fällig' (due) is often combined with obligations. Typical nouns are 'die Rechnung' (the bill), 'die Miete' (the rent), 'die Zahlung' (the payment), or 'der Beitrag' (the contribution/fee).
Der jährliche Mitgliedsbeitrag für den Sportverein ist fällig.
The annual membership fee for the sports club is due.
When it comes to expected events, 'fällig' (due) is often connected with regular appointments or maintenance. Common nouns are 'die Inspektion' (the inspection), 'die Prüfung' (the exam/check), or 'die Untersuchung' (the examination/check-up).
Die jährliche zahnärztliche Untersuchung ist mal wieder fällig.
The annual dental check-up is due again.
In this meaning, 'fällig sein' (to be due) is colloquial and expresses that a consequence or action is logical or deserved. It can be used for both positive and negative things.
Nach der harten Arbeit ist jetzt wirklich eine Pause fällig.
After the hard work, a break is really due now.
necessary
Both words can refer to a break that is needed after a long period of work. „Nötig“ describes the existing need, while „fällig“ emphasizes that the break is now due or expected. Use „nötig“ for the need and „fällig“ for the timing.

Nach drei Stunden Arbeit ist jetzt eine kurze Pause fällig.
After three hours of work, a short break is now due.
Nach drei Stunden Arbeit ist eine kurze Pause nötig.
After three hours of work, a short break is necessary.