förmlich
formal
Stiff, official, and impersonal in accordance with social conventions.
Die Begrüßung durch den Direktor war sehr förmlich und distanziert.
The greeting from the director was very formal and distant.

formal / practically
Loading your saved state.
formal
Stiff, official, and impersonal in accordance with social conventions.
Die Begrüßung durch den Direktor war sehr förmlich und distanziert.
The greeting from the director was very formal and distant.

The word is derived from 'die Form' (form). Something that is 'förmlich' (formal) follows strict social forms or rules.
Bei dem Staatsbesuch ging es sehr förmlich zu.
The state visit was conducted in a very formal manner.
If 'förmlich' describes a person or situation as stiff and official, the first meaning is intended. If it intensifies a verb or adjective in the sense of 'geradezu' (practically), the second meaning is intended.
Die Atmosphäre bei dem Treffen war eher förmlich.
The atmosphere at the meeting was rather formal.
In the meaning 'offiziell' (official), the word is often used for communication and behavior: 'eine förmliche Anrede' (a formal form of address, e.g. 'Sehr geehrte Damen und Herren' — Dear Sir or Madam), 'ein förmliches Schreiben' (a formal letter), or 'eine förmliche Einladung' (a formal invitation).
In Geschäftsbriefen ist eine förmliche Anrede üblich.
In business letters, a formal form of address is common.
In its intensifying meaning, 'förmlich' comes directly before the word it emphasizes. It often describes a strong, almost exaggerated reaction or state.
Nach der langen Wanderung war ich förmlich ausgehungert.
After the long hike, I was practically starving.
official
'offiziell' and 'förmlich' can both describe stiff, impersonal interaction. 'förmlich' fits especially well when talking about language, clothing, politeness, or social rules. 'offiziell' can sound that way too, but it often additionally carries the idea of an office, function, or public role. Learners may confuse the words because an official occasion is often also formal. In practice: if you mean the style, 'förmlich' is more precise. If you mean the official or work-related side, 'offiziell' fits better.

Im Brief an die Ärztin wählt Mehmet eine förmliche Anrede.
In the letter to the doctor, Mehmet chooses a formal form of address.
Der Empfang beim Bürgermeister war sehr offiziell.
The reception at the mayor's office was very official.