The tides would be much more complex and perhaps less predictable. If the two moons were on the same side of the earth, the tides would be very high. If they were on opposite sides, the tides would be smaller—how much smaller depends on the relative masses of the moons. Anyway, you could probably write any kind of weird tide you wanted with a double moon system.
There there’s the light at night. Apparently, tomatoes ripen more under a full moon. Without a full moon, they won’t ripen as much. With two moons, each being full at different times, you might have a much different tomato yield.
And tomatoes aren’t the only flora (or indeed, fauna) that are affected by the light of the moon. You’d have to research it, but I’m sure you’d find a lot of interesting moonlight effects.
If your story is a bit magical, then imagine the werewolf on a double full moon! Or what would happen to vampires? Would their movement be inhibited on double full moons nearly as much as during the day? Would this make them even more frantically active on double new moons? They myths about double night, as it might be called, could be an interesting thing to play with.
Have you ever read Asimov’s “Nightfall?” If not, you should. The planet is in a six moon system where full dark night happens once every two thousand years. This has a profound effect on the civilization, as everyone freaks out and goes absolutely apeshit each time it happens. So much so, that civilization falls, and has to be rebuilt over the next 2000 years, only to be destroyed again.
And the moons could not always be opposite each other, unless they were the same mass and somehow were stable in opposite orbits. I’m not sure, but I don’t think that would be possible. There’s something called the “three body problem.” I’m not sure what it is, but since they are always looking for a solution, it seems to me that it might be something about how impossible it is to solve such a problem. Maybe you can never predict the orbits in a system with three bodies.
Perhaps the culture knows this, and is constantly wondering when doom will fall, as the system destroys itself.
[Edit]
Yup: here’s what wikipedia has to say about the three body problem. It seems to me that such systems (which are everywhere) are inherently unstable. Your planet would probably be worse off than us when we consider the Earth, the Moon, and the Sun.