
Vollendung
completion
The act of finishing something or the successful conclusion of a process, plan, or work.
Die Vollendung des Bauprojekts wird noch mehrere Monate dauern.
The completion of the construction project will still take several months.
completion / perfection
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completion
The act of finishing something or the successful conclusion of a process, plan, or work.
Die Vollendung des Bauprojekts wird noch mehrere Monate dauern.
The completion of the construction project will still take several months.

perfection
The state of the highest quality, perfection, or flawlessness.
Sie spielte das Klavierstück mit technischer Vollendung und großem Gefühl.
She played the piano piece with technical perfection and great feeling.
'Vollendung' (completion/perfection) is derived from the verb 'vollenden' (to complete), which consists of 'voll' (full) and 'enden' (to end). It literally describes the act of bringing something to a 'full end,' that is, completing it fully.
Der Künstler arbeitete jahrelang an der Vollendung seines Meisterwerks.
The artist worked for years on the completion of his masterpiece.
Common phrases connected with the completion of a process are 'etwas zur Vollendung bringen' (to bring something to completion) or 'vor der Vollendung stehen' (to be nearing completion).
Der Architekt will das Gebäude noch in diesem Jahr zur Vollendung bringen.
The architect wants to bring the building to completion this year.
To express the highest perfection, 'Vollendung' (perfection) is often used with the preposition 'in' or 'mit' (with). The phrase 'in Vollendung' then means something like 'perfectly' or 'to perfection.'
Er beherrscht die Kunst des Geigenspiels in Vollendung.
He has mastered the art of violin playing to perfection.
completion / degree
Abschluss and Vollendung can both express that a work or process is finished. Abschluss mainly refers to the endpoint. Vollendung places more emphasis on the planned work having been successfully completed and therefore sounds more ceremonious.

Nach der Vollendung des großen Wandbilds lud die Künstlerin zur Feier ein.
After completing the large mural, the artist invited people to a celebration.
Nach dem Abschluss der Arbeiten räumte die Künstlerin das Atelier auf.
After completing the work, the artist tidied up the studio.