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You can see that this antenna has fairly good Front to Back Ratio (FBR), which means that most of the radiation is focused forward and not much is focused backward. Now, if we were able to get more gain out of this antenna, you would see the main lobe less wide than it used to be; in addition to a farther radiation distance.
The short answer was provided above - more directivity is more gain. Be careful though. A beam antenna can be quite "directive" and direct as much power forward as backward - or in other specific and predictable directions (lobes if you are looking at a plot of the pattern). Front to back ratio does play a part. The more focused the power in a specific direction (don't forget horizontal and vertical) the more gain.
While directivity is the ratio of power in a particular direction vs total power radiated, gain is a similar ratio but instead of total power radiated, the denominator is the total power delivered to the antenna. So, the difference between the two terms is that power gain takes into account the losses in the antenna plus the mismatch losses at the interface between the antenna and the transmission line, while directivity does not. If there are no losses and the match between antenna and feedline is perfect, then directivity equals gain. In practical cases the gain is always less than the directivity because there are always some degree of losses.
Both power gain and directivity are a function of direction, but convention says that if there is no direction stated, then the gain or directivity figure given is that in the direction of maximum gain. For clarification, this is sometimes stated as the peak directivity or peak gain. If a direction is stated, then there is no ambiguity.