erwünscht
welcome
Describes something that is wanted, welcome, or hoped for. It expresses that the presence or occurrence of something is viewed positively.
Bei diesem Projekt ist jede Hilfe erwünscht.
Any help is welcome with this project.

desired / welcome
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welcome
Describes something that is wanted, welcome, or hoped for. It expresses that the presence or occurrence of something is viewed positively.
Bei diesem Projekt ist jede Hilfe erwünscht.
Any help is welcome with this project.

'Erwünscht' (desired/welcome) is the past participle of the verb 'erwünschen' (to desire). The verb 'wünschen' (to wish) is intensified by the prefix 'er-'. As an adjective, it therefore describes something that is strongly or officially desired.
Der erwünschte Erfolg blieb leider aus.
The desired success unfortunately failed to materialize.
This adjective is very often used predicatively with the verb 'sein' (to be). The structure is then: 'Etwas ist (nicht) erwünscht.' (Something is (not) desired/welcome.)
Ein offener Dialog ist in unserer Firma ausdrücklich erwünscht.
Open dialogue is expressly welcome in our company.
In the meaning 'gesellschaftlich passend' (socially appropriate), 'erwünscht' (acceptable/desired) is almost always negated. The phrase 'nicht erwünscht' (not desired/not allowed) is often found on signs or in house rules to make rules clear.
Das Mitbringen von eigenen Speisen und Getränken ist nicht erwünscht.
Bringing your own food and drinks is not allowed.
When something is welcome, 'erwünscht' (welcome/desired) is often combined with positive, abstract nouns. Typical phrases are 'Hilfe ist erwünscht' (help is welcome), 'Vorschläge sind erwünscht' (suggestions are welcome), or 'das erwünschte Ergebnis' (the desired result).
Für die Planung des Festes sind kreative Vorschläge jederzeit erwünscht.
Creative suggestions for planning the celebration are welcome at any time.
welcome
'willkommen' and 'erwünscht' can both say that something is viewed positively or is wanted. 'willkommen' often sounds warmer and more personal, especially with guests or help. 'erwünscht' sounds more matter-of-fact and is often used in rules, notices, or official situations. It can also say what behavior is appropriate. Learners may confuse the words because both express 'not against it, but in favor of it.' In invitations and friendly situations, 'willkommen' is usually more natural; in rules such as 'Anmeldung erwünscht' (registration desired), 'erwünscht' fits better.

Bei der Bewerbung ist Erfahrung im Verkauf erwünscht.
Experience in sales is desired for the application.
Zum Nachbarschaftsfest sind auch Kinder herzlich willkommen.
Children are also warmly welcome at the neighborhood festival.