I am trying to recreate this, and I have been fairly successful. I am having issues with the collision handling though. Although the collision handling seems to work, it has very strange behavior. Here is what I have so far. This is the code that handles collisions:

var dx = particle2.getX() - particle1.getX();
var dy = particle2.getY() - particle1.getY();
var angle = Math.atan2(dy, dx);

var newP2X = particle1.getX() + (particle1.getRadius() + particle2.getRadius()) * Math.cos(angle);
var newP2Y = particle1.getY() + (particle1.getRadius() + particle2.getRadius()) * Math.sin(angle);

particle2.setX(newP2X);
particle2.setY(newP2Y);     

var p1Vxi = particle1.getVx();
var p1Vyi = particle1.getVy();
var p1Mass = particle1.getMass();

var p2Vxi = particle2.getVx();
var p2Vyi = particle2.getVy();
var p2Mass = particle2.getMass();

var vxf = (p1Mass * p1Vxi + p2Mass * p2Vxi) / (p1Mass + p2Mass);
var vyf = (p1Mass * p1Vyi + p2Mass * p2Vyi) / (p1Mass + p2Mass);

particle1.setVx(vxf);
particle1.setVy(vyf);
particle2.setVx(vxf);
particle2.setVy(vyf);

EDIT: I have tried to change it to inelastic collisions like suggested, but for some reason the balls collide erratically. Check it out here.

Any help is much appreciated!