Antenna effiency in bluetooth module
In broad terms, the efficiency of the antenna is the measure of how efficiently it radiates energy. For example, if an antenna is 50% efficient and we put in 1W of electrical energy, it will radiate (turn into RF radiation) 0.5W of RF energy. Obviously a higher efficiency is good, because it means we get more RF energy.

Antenna gain is a bit more complicated. Basically, different antenna shapes will have different radiation patterns. For example, with a directional antenna, it will radiate a lot of energy in a thin beam in the direction that it's pointing, and not much energy in any other direction. This gain is relative to the radiation pattern of an isotropic antenna, which is a non-directional antenna that radiates equally in all directions. The gain can be measured in different directions from the antenna, but usually when only a single number is given, it refers to the peak gain, i.e. how much better is the antenna at transmitting in a specific direction than an isotropic antenna.

More antenna efficiency is always good, but more gain is only good if you want more directionality. This may not always be the case, for example if you dont know where the other antenna is, you would want power to be radiated more evenly in all directions.

In this case, the manufacturer is quoting a distance for the module and the antenna together, so you can assume that the range is 100m in at least some directions. However, depending on the directionality of the antenna, the range may be lower in other directions. If you removed that antenna and put on an antenna with a higher efficiency, like 100% and the same gain, the range would be 142m (in an ideal situation).