Difference between directional and beamforming antennas
I would say there is some overlap, but a 'directional antenna' is usually going to have a fairly simple transmit and receive pattern aimed roughly in a particular direction -- it will likely not be uniform, but will be stronger in the center and fall off towards the edges -- whereas 'beamforming' normally refers to more clever tricks to carefully control precisely where the beam goes by varying the relative phase and amplitude of multiple transmitting/receiving elements.

If one makes a directional antenna using multiple active elements with a specific fixed phase shift between them, arguably one is doing a very simple kind of beamforming. But normally the word is reserved for larger and more complex arrays with more detailed control over the beam, up to and including active systems that use software control over the shape and direction of the beam in response to conditions.