Consider an omnidirectional antenna with X dBi gain. Intuitively I understand that the power that would have been dispersed in now-dead directions is directed around the antenna.
What I don't get is that we find Yagi Uda antennas with gains around 10dB, but also omnidirectional antennas of the same gains.

What's the point in getting a Yagi Uda over an omnidirectional if it has the same gain? I thought directional antennas would redistribute the power from one direction to a privileged direction, so shouldn't we end up basically with a total gain of that of an equivalent omnidirectional + the directivity?

Otherwise I would expect a Yagi Uda of gain X to have a larger vertical beamwidth than an omnidirectional of the same gain, but it does not seem to be the case so I am puzzled.