The situation here is unsurprisingly pretty complex.
There's a lot of fairly well-substantiated anxiety that we're going to be next in line due to not being in NATO. Our Foreign Policy Institute said that it's fairly likely Russia will be using military force against us in the "next few years". This really hammers home the sort of anxiety we're talking about here.
A couple of years ago the majority of Finns were against joining, in recent polls it's been about 50/50. It definitely says something that it's still only 50/50, though.
The main argument against NATO seems to revolve around our supposed historical neutrality, but in all honestly it feels more like Finlandization than an actual argument; our "neutrality" has never been exactly neutral, starting with being allied with the Nazis (totally only for convenience, we swear, no ideological ties whatsoever, no sir). Another (and IMO more credible) argument is that we don't want to be dragged into a war by the US – people don't really trust the US especially after Trump, and the US's human rights abuses are often pointed out in NATO discussions. Some also feel a bit iffy about joining an alliance with Turkey, whose human rights record is also not exactly what you'd call stellar. Edit: many especially on the left (where most of the resistance to joining NATO is coming from, in addition to some of the ultranationalist nutballs who love Putin) also say that joining a military alliance makes no sense if you want peace.
The debate you're referring to probably meant the various citizens' initiatives that popped up. We have an online initiative system where the parliament has to consider every initiative with over 50k signatures (done using an eID provider). They, however, aren't in any way compelled to do anything about them, and the majority of initiatives really don't lead to squat unless there was existing political will to do it in the first place.
I'd give about 50-70% odds of us going for NATO membership, although the question is how do we do that without Russia eg. starting a border skirmish before the membership ratification round is done.