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I will speak for Argentina, in case it is different in other Latin American countries or in Spain.
What we say to anybody on their birthday is "feliz cumpleaños". I gave "Feliz Cumpleaños" in my answer because that is how it is usually written on a card. So, "feliz cumpleaños" is the standard oral form. In a card you can have more elaborate forms: "Te deseo un feliz cumpleaños" or "Te deseo un muy feliz cumpleaños". You can also say "Que pases un feliz cumpleaños", or "Que tengas un cumpleaños muy feliz". The "Happy Birthday" song in Spanish has two lyrics that I know of, starting: "Que los cumplas feliz..." or "Cumpleaños feliz, te deseamos a ti...".
I think "feliz cumpleaños" (probably with the first letter of each word in upper case if it's a card or letter) is what you are looking for.
About "Felicito con cumpleacos" or "Te felicito con tu cumpleacos": "cumpleacos" is "cumpleaños" misspelled; "Te felicito por tu cumpleaños" is a possibility if you are writing, but I doubt you would hear it in a conversation, it sounds too formal.
>And what's the difference between "cumpleacos" and "cumpleaños"?
Spelling! The proper spelling is "cumpleaños". If you have problems finding the "ñ" in your keyboard you can type Alt + 164.