An open-ended spanner only engages on two faces of the nut. As you can see in the picture below, the inside of the spanner slot is not shaped to engage with the nut. Open-ended spanners are used where the only possible access to the nut is pushing it across the faces of the nut. The "play" between the spanner and the nut means that the spanner exerts most force on the corners of the nut, which risks rounding the nut off.
A flare nut spanner engages on four faces of the nut, as you can see from the picture below. The spanner has lips at the front which ensure it engages on the inside of the spanner slot. Most of the force is still born by the two opposing faces, but with well-defined engagement with the spanner it is possible to shape the spanner so that it contacts the nut more towards the centre of the nut faces with a wider contact area, in the same way as a good box wrench does, instead of on the corners of the nut as an open-ended spanner does.
Of course the lips on the spanner mean you can't just slide it onto the nut, as you would for a regular open-ended spanner. For plumbing/hydraulic applications, the nut is (almost) always wider than the connected pipe, so you slot the spanner over the thinner pipe and then push it down onto the nut.