Hey,
There is a very interesting scientific process that leads to the formation of tiny water droplets on a cold water bottle and it is called 'Condensation'.
The ice water is at zero degree Celsius or slightly more than it. For condensation to occur,the dewpoint has to be above this. The air surrounding the bottle of cold water is cooled closed to freezing point.Moisture in the air will condense out once the temperature of the air drops below the initial dewpoint of the air. The higher the dewpoint then the greater the amount of condensation that will occur over a given time. Thus, overall humid conditions (high dewpoint, high relative humidity) will lead to a greater buildup of moisture over a given time on the glass which will appear in form of droplets.
Hope this helps.
Good luck.