Veitsi, terä, and (less commonly) puukko all seem to be used. Terä seems to be a generic "blade" including the cutting head of an axe, or the descriptor of the edge - veitsenterällä. Puukko seems to be a sheathed knife. Piikkari and tikari seem to be specific to things like daggers and stiletto knives. Hopefully someone else will jump in with more specifics!
terä cannot be used as a synecdoche for the whole knife/sword in Finnish the same way "blade" works in English. It always refers only to the sharp edge. puukko is usually reserved for traditional Finnish sheathed knives (although many people call cheap Mora knives puukko too). piikkari is actually a running shoe for track & field, so it has nothing to do with knives. A dagger is indeed called tikari, but a switchblade is usually called stiletti. A machete is viidakkoveitsi, literally "jungle knife". A small pocket knife is called linkkuveitsi in Finnish. :)