Well, Vigo was a sorcerer who painted a self-portrait, then enchanted it so that when he died, his spirit would inhabit/haunt it. The spell draws strength from negative human emotions, so that over time, it turns them into ectoplasm, and when enough of the slime accumulates and flows to him, he can take form outside the painting or possess someone and make them his new body.
The guys defeated him first by countering all the city's negative emotions with positive ones via New Year's Day celebrations. Then they covered him in his own mood slime, charged with positive emotions instead of negative and forced him back into the painting with their proton streams.
When they did this, the painting depicted him exploding and the portrait changed to depict the Ghostbusters and Oscar.
Those are the facts from the end of GB2.
Now…as for why he's back: Energy (including ghosts) can neither be created nor destroyed. That means that explosion was just a dramatic display on the part of the enchanted paint and Vigo's ghost is still trapped inside the painting. After all the New Year's celebrating ended and New York went back to being miserable, it makes sense that the paint would change to reflect Vigo once again…albeit he's much weaker now with the river of slime gone.
And then the guys apparently stole the painting from the museum, judging from one of the answering machine messages, to keep a safe eye on it and study it.
As for the new equipment, well…in the EGB continuity, the city gets so quiet that the Ghostbusters run out of ghosts to bust and go out of business once again, not reopening until 1997. Obviously, all the equipment would've been packed away at that time. The reason it's never been brought out of storage since is, if you notice in EGB, they almost never bill anyone and the firehouse and Egon always look like a shabby mess, so obviously they weren't making much money…and obviously that Dark Matter and Meson tech is extremely costly to build, maintain, and operate, and while useful, isn't strictly necessary to do the job.
I wouldn't count on it staying in storage much longer, though. Several of us over at the West Coast franchise are eager to start writing it in.