Yes, it's different. What they're trying to prevent is the water from the reactor itself from being released into the atmosphere. That's the water that comes in direct contact with the nuclear fuel. This water is radioactive because the steam carries a lot of radioactive particles in it. You don't want to release a lot of that into that atmosphere.
They're most likely worried about the reactor and the primary containment around the reactor being compromised. If primary containment breaks in a reactor building with walls already broken, the only thing separating the radioactive steam from the atmosphere is a couple of inches of steel. If the reactor walls themselves break, then you have a leak of radioactive steam/water. And since a reactor is essentially a giant pressure cooker (reactors operate around 1020+ psi), a substantial crack or break can be very bad.
The FAST way to relieve pressure would be to just vent all the steam really quickly; either directly into the atmosphere or though a ton of heat exchangers. The problem with that is that you end up releasing a lot of radiation, so on the scale they'd need to do it here, they can't do it.
What boric acid and water do is kill and cool the fuel. Boron kills nuclear reactions; that's what they use on control rods. It stops the fuel from reacting and generating heat. Water pumped through a reactor draws away heat. So what they're doing is reducing the heat in the reactor so that they can relieve the pressure.
The thing is, once you dump boric acid into a nuclear reactor, you have basically killed your reactor. You can never use it again, unless you spend an absolutely exorbitant amount of money to clean it. This is a big deal. Obviously safety is a concern and everyone wants to prevent a release of radiation, but nuke plants generate a lot of electricity and they're expensive to make. So dumping the liquid poison (i.e. boric acid) into a reactor is the absolute last resort.
This is different from "nuclear waste getting into the water supply." It's not really waste they're worried about in a meltdown. It's the nuclear fuel that's still reacting and very very hot. If they can't control the temperature of the fuel (with cooling water and if necessary, boric acid), it just keeps reacting and getting hotter until it melts through the reactor itself; it will continue to melt through down into the earth and could possibly come in contact with groundwater, which is very very bad.
They are two separate issues.