Which is correct, and why?:
Growing my business has been a welcomed challenge.
OR
Growing my business has been a welcome challenge.
Which is correct, and why?:
Growing my business has been a welcomed challenge.
OR
Growing my business has been a welcome challenge.
You should use the second form. They're both correct, but only because 'welcome' is a funny word.
When you slime something, it becomes slimed.
When you pay something, it becomes paid.
When you welcome something, it becomes welcome!
Welcoming something imbues it with an fresh identity. It is now a guest. It is welcome. This is more serious than just saying something is now Xed because you did X to it.
The first form you provided is clearly a passive expression. It sounds like you are avoiding taking responsibility for the welcoming. The second form is not clearly a passive expression. It's idiomatic, anyway, and people will read it as you personally welcoming the challenge. Since the welcome in that case has become an intrinsic attribute of the challenge, the reader can safely assume it was you who granted it that status. With the first form, that assumption is not safe.
If you wish to be more bold you could say 'I have welcomed the challenge of growing my business'
Nouns and adjectives need not be modified using the participle form.
welcome n., v., adj.
ADJ. 2 Very pleasing because much needed or desired:
after your walk, the tea room serves a welcome cuppa
Unlike certain other verbs, the past-participle (ed) form is not needed in case of welcome, as it is essentially a noun and therefore can be used directly in an adjectival sense. Participial adjective forms are used only with some verbs.
The two words are almost the same in meaning, and yes, it is because "welcome" is a funny word. Funny, because it already is, in a sense, a past participle, at least etymologically. Part of the derivation of "welcome" is this: "Middle English < Scandinavian; compare Old Norse velkominn, equivalent to vel well1+ kominn come (past participle)" (from Dictionary.com) Something like "You have arrived well." "Welcome" would be the most versatile choice, use "welcomed" when emphasizing the act of those who did the welcoming: "It was a welcome reminder that we live in an enlightened age" vs. "We were welcomed with a barrage of angry catcalls".