As an enabler, I would say yes, YES, you need a boar brush. From a practical standpoint, however, no you don't. Chris hit the nail on the head in his description the difference between the different brushes.
The only thing I would add, is that to me, there are three things (outside of where they came from and appearance) that differentiate the brush types.
Face Feel: Each brush type is going to feel quite different on the face compared to one another. I have a boar brush (a Semogue 830) that is actually softer on the face than any of my badger or synthetics. Synthetics are going to be the softest when dry (how soft varies between the type of synthetic), but typically will have more scrub/scritch when face lathering due to the corregations in the fibers. Badgers might feel very scratchy when dry, but will soften up quite a bit when soaked. The hair shafts are smooth, so when face lathering tend to feel more luxurious when combined with soft tips. Boar brushes need quite a bit of break in time to reach their potential. When they get there, they will have the most backbone out of any brush when dry. Depending on the knot type, that backbone can be lost when soaked, but the tips are extremely soft. The hairs, unlike badger, will split on the ends.
Density: In my experience, you're going to find that badger hair brushes can be packed much more densely than the other two brush types. This will provide additional backbone to the brush. Synthetics are generally next in the density category, followed up by boar.
Water and heat retention: All three brushes release water differently. Truth be told, this is where many people develop an affinity to one brush type or another. Badger brushes release water very slow, while synthetics are the opposite, they release what water is held in the knot almost immediately. Boar, from my experience, is somewhere in between. As far as heat retention, there's no contest. Badger has been shown to hold heat 30% longer than synthetics, and 50% longer than boar. This is mainly down to the density of the knots, with the hair type playing less of a role. If you had an extremely dense boar knot, I'm sure it would hold heat longer, though I'm not sure it would be able to keep up with the badger.
As pointed out, both boar and badger hair knots need to be soaked (badger for at least 30 seconds to a minute, boar 2-3 minutes), while synthetics do not since they don't absorb water.
Synthetics and boar are outstanding value for what you get, however...my preference is for badger and synthetic, but I'll always keep my Semogue around.
