Directivity compares radiation in a direction with an isotropic radiator using the same total radiated power. Gain includes dissipative efficiency; realized gain also includes impedance mismatch at the declared feed. Beamwidth describes an angular width of the main beam, commonly between half-power points, while sidelobes are local radiation maxima outside that main lobe.
A high peak gain does not guarantee adequate coverage. The service region may need a minimum realized-gain envelope, ripple limit, sidelobe ceiling, null depth, front-to-back ratio and cross-polar limit over frequency. Beamwidth and gain also trade against each other for a given effective aperture.
Ask which gain quantity is reported, at which frequency and polarization, in which coordinate system, with what normalization and measurement uncertainty. Use the full pattern grid for system analysis rather than accepting one boresight value.