Bienvenue vs. bienvenu
The main difference between the French words bienvenue and bienvenu depends on their grammatical role. The word bienvenue is used as an invariable greeting or a feminine noun. Conversely, bienvenu is used as an adjective or a noun describing the person or thing being welcomed, and it must agree in gender and number with the subject.
Bienvenue
When used as an exclamation to greet someone upon arrival, bienvenue is invariable and always takes the feminine singular form regardless of who is being addressed. It is also used as a feminine noun to refer to the act of welcoming someone, commonly found in phrases about wishing someone a warm welcome.
Bienvenue dans notre nouvelle maison.
(Welcome to our new home.)
Je vous souhaite la bienvenue au nom de toute l'entreprise.
(I welcome you on behalf of the whole company.)
Bienvenue en France et bon séjour.
(Welcome to France and enjoy your stay.)
Le directeur a prononcé un petit mot de bienvenue.
(The director gave a short welcome speech.)
Bienvenue sur mon site internet.
(Welcome to my website.)
Nous avons organisé une fête de bienvenue pour les nouveaux étudiants.
(We organized a welcome party for the new students.)
Bienvenu
The word bienvenu functions as an adjective or a pronoun used to describe someone or something that is received with pleasure. Because it modifies a subject, it must change its spelling to match the gender and number of that subject, becoming bienvenu for masculine singular, bienvenue for feminine singular, bienvenus for masculine plural, and bienvenues for feminine plural.
Tu es toujours le bienvenu chez moi.
(You are always welcome at my place.)
Si tu veux te joindre à nous, tu es la bienvenue.
(If you want to join us, you are welcome.)
Vos commentaires et suggestions sont les bienvenus.
(Your comments and suggestions are welcome.)
Cette pause café est vraiment la bienvenue après une longue réunion.
(This coffee break is really welcome after a long meeting.)
Chers amis, vous êtes les bienvenus à notre mariage.
(Dear friends, you are welcome at our wedding.)
Toutes les nouvelles idées seront les bienvenues.
(All new ideas will be welcome.)
Summary
In short, use the invariable bienvenue when you are directly greeting someone upon arrival or referring to the noun of welcoming. Use bienvenu when you are describing a subject that is welcome, making sure to adapt the spelling to match the gender and number of that specific subject.