LLM is mostly correct. When talking about traditions using "tavat" is mostly something along the lines "This is the way we've always done this" or "this is a common habit" while "perinne" is more literal translation of "tradition" or "heritage". "Perinne" is originally used when talking about folklores or older ways of doing things but it's also common to create your own traditions and "perinne" has a bit deeper and cultural meaning than just "tapa".
Then there's things like "perinneruoka" (traditional food), "perinneyhdistys" (society to preserve and keep alive knowledge about traditions and their backgrounds related to something) and "perinnerakentaminen" (traditional way of building stuff) where "perinne" clearly refers to older stuff with cultural background.
"Tapa" is more clearly a habit or custom and it usually means on doing something. You can even say "Olen ottanut tavaksi noudattaa perinteitä" (I've taken a habit of following traditions). 'Manners' would literally translate to 'käytöstavat' (behavior) where you're for example expected to hold the door for someone behind you out of courtesy and while it could be explained as 'traditional behavior' the actual act is rarely called 'perinne'.
Hope this helps.