behördlich
official
Coming from an authority or relating to an authority; official, administrative.
Für den Bau des Hauses ist eine behördliche Genehmigung erforderlich.
An official permit is required for building the house.

official / administrative
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official
Coming from an authority or relating to an authority; official, administrative.
Für den Bau des Hauses ist eine behördliche Genehmigung erforderlich.
An official permit is required for building the house.

The adjective 'behördlich' (official/administrative) is derived directly from the noun 'die Behörde' (authority, government office). The ending '-lich' is often used to form adjectives from nouns that describe belonging or a characteristic.
Sie müssen das Formular bei der zuständigen Behörde einreichen.
You must submit the form to the responsible authority.
This word belongs to formal official and administrative language. In normal everyday language, people would more likely say 'offiziell' (official) or 'amtlich' (official), but in documents and official letters, 'behördlich' is very common.
Der Bescheid wurde Ihnen auf dem behördlichen Postweg zugestellt.
The notice was delivered to you through official postal channels.
'Behördlich' is almost always combined with nouns from the fields of administration and law. Common combinations are 'behördliche Anordnung' (official order), 'behördliche Auflage' (official requirement), or 'behördliche Vorschrift' (official regulation).
Das Unternehmen musste neue behördliche Auflagen erfüllen.
The company had to meet new official requirements.
public
'öffentlich' and 'behördlich' can both have to do with the state, administration, or the municipality. 'öffentlich' refers more to the general community or the public sector, for example public schools, public transportation, or public institutions. 'behördlich' is more precise: something comes from an authority or relates to an authority. The words can be confused because authorities are part of the public sector. But not everything public is 'behördlich'. 'behördlich' fits especially with permits, letters, inspections, or orders.

Für den Umbau braucht die Familie eine behördliche Genehmigung.
The family needs official approval for the renovation.
Viele Menschen fahren morgens mit öffentlichen Verkehrsmitteln zur Arbeit.
Many people take public transportation to work in the morning.