How to use the Finnish words 'mutta' and 'vaan'

What's the difference between these two buts?

Mutta

• Laitoin herätyskellon soimaan, mutta nukuin silti pommiin. - I set the alarm clock, but I still slept in.
• Haluaisin tulla, mutta minulla on kokous samaan aikaan. - I'd like to come, but I have a meeting at the same time.
• En pääse, mutta olen hengessä mukana. - I cannot make it, but I'll be with you in spirit.  

Vaan

• En tilannut pihviä vaan salaatin. - I didn't order a steak but a salad.
• Ei se ole minun vaan Kallen! - It's not mine but Kalle's!
• Täällä ei juhlita vaan mennään aikaisin nukkumaan. - Here we don't party but go to sleep early.

So, usually mutta starts an explaining sentence of its own, and vaan gives an alternative, something else instead.

Notice that vaan is also the spoken language form of vain, which means only, just.

• Tää on vaan tytöille! - This is only for girls!
• Miks sä ostit vaan kaks perunaa? - Why did you buy only two potatoes?
• Joo, ota vaan se tuoli. - Yeah, go ahead and take the chair.
• Mene vaan. - Just go.
• Read more about spoken Finnish here.

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– Hanna