Its effect would be immense.
Modern armor needs to do two things: prevent penetration and spread the force of impact. Most of our current body armor is focused on preventing penetration. Composite armors are just that: several layers of materials with a specific purpose to stop a bullet.
The first layer is often a cloth or Kevlar layer that holds the composite material.
The second layer is a hard material designed to absorb the most impact and spread the force. Such hard materials are hard but brittle and shatter while doing its job. That isn't ideal against repeat hits but it works. This is also why just dropping the composite plates on the ground a few times can already degrade the quality enough that you might have to discard the entire armor plate, which seems counter-intuitive for a material supposed to stop a bullet.
The last layer that is always there is a softer more ductile material. This is designed to "catch" the bullet and any shrapnel created by the shattering of the hard layers. As this layer catches the material the force of the bullet is transferred to the body. If the topmost materials have done their job this has spread the force across a larger area on your body.
With that basic understanding of modern body armor you can see that most of the problems are centered around preventing penetration. Your magic material erases that problem and allows you to focus on spreading the force.
If you drape the material over two studs a few millimeters away from the surface below (basic armor plates are between 8 to 25mm thick before you add the Kevlar and shirts and whatever else is worn to it), any force between those studs is spread between them. Make sure the material doesn't have the elasticity to bent far enough and hit the body and you are set.
Now create a studded body armor of a lightweight material and stretch the magic material over the studs. The studs are extremely squad, having a small point at the top and broad at the base. Some room needs to be left so you can still bend and move. Any bullet landing anywhere on the armor will A: stretch the magic material and slow down and B: spread the force over the studs, who in turn spread it over the body (or spread it over a harness beneath that before it really touches the body). You use at least two layers of this, for example with 10mm high studs for a thickness of 20mm of this armor.
Q: but won't the studs reduce maneuverability?
A: it might reduce maneuverability, but never as much as a modern solid armor plate does right now. It also offers more coverage under the arms for example. Studs can also have more space in between them at places where more maneuverability is needed to still provide protection. Ergonomic designs would mean you then increase the amount of studs nearby places that don't need as much maneuverability and can handle the shock. Since joints for example are usually also more susceptible to damage it reduces the chance of incapacitation. Example for the spine: studs are placed over the ribs on either side, making sure that wherever the bullet hits at least 4 studs are involved in taking the force.
Q: the material needs to stretch when you move, a few millimeters movement would mean you can barely move!
A: The distance between the studs is important here. If the distance between the stud tips is less than the maximum elasticity in mm when your movement causes those tips to be furthest apart you have no repercussions to your movement. It is likely that in some places you need more distance between the stud tips, at those places you can increase the distance between the studs while increasing the height of the studs to more than 10mm.
Q: with such a small surface area between the tips a bullet would just ram the studs through your body!
A: even if a bullet lands perfectly in the middle of two studs, any studs to the sides will also help. On top of that the studs could be placed on their own material to spread the force. An example of the two layer setup: one layer of magic material, thin studs that support the magic material, another layer of magic material the studs rest on and then another layer of studs but with a thicker base. Each individual stud in the first layer would be placed exactly between four studs on the second layer, meaning a perfectly placed bullet would be spread over a minimum of 8 studs in the second layer ignoring the fact that a portion of the force would be carried by two more topside studs to the side.
Q: what if I hit a stud dead-on? Now you are dead!
A: the studs would likely be made of the magic material itself, but rolled up and compressed to create hardness. If you hit a stud dead-on it would be suppressed and cause the studs around it to take a portion of the force. Even if that somehow doesn't happen you still spread the force over 4 studs below. If the magic material isn't useful enough to act as studs and you use a metal instead, then you'll likely lose that stud. However your opponents would need to break a lot of studs by hitting them dead-on before the armor degrades enough to falter, especially if you go for more than 2 layers.
On top of all of that: shields are useful again! A handheld shield with a similar setup would easily help protect that person. And you would naturally use a few hundred layers of this stuff on tanks and other vehicles. It would instantly mean that any weapon you bring needs to be a lot bigger than before. The unfortunate consequence is that alternatives to kinetic projectiles will be sought. Chemical warfare would likely rise up quickly.