Vapor pressure refers to only the partial pressure of the gaseous form of your liquid. In a closed system at equilibrium (e.g. if you seal a beaker containing water), the partial pressure of water vapor above the liquid will be the vapor pressure of water regardless of the partial pressures of other gases in the air (e.g. N2, O2).
The boiling point of a liquid, however, does depend on atmospheric pressure. Why is this? Well, as I mentioned before, the vapor pressure is essentially the Kp for the reaction H2O (l) <--> H2O (g). If this is the case, why does vaporization take place only at the surface of water? For other chemical reactions, the reaction will take place all throughout the solution and not only at the surface.
The answer has to do with pressure. When the vapor pressure of a gas exceeds the atmospheric pressure, the vaporization of water does occur all throughout the solution (not entirely true, more on this later), creating a phenomenon known as boiling. Boiling happens only when vapor pressure exceeds atmospheric pressure because vaporization in the middle of the solution requires the formation of bubbles that have an internal pressure large enough to counteract the external pressures that try to squeeze the bubble back together (roughly atmospheric pressure). Since the interior of the bubble is entirely vapor, the highest pressure inside the bubble is the vapor pressure of the liquid. So, when the vapor pressure is smaller than atmospheric temperature, any bubbles are immediately crushed back into solution by atmospheric pressure and evaporation can occur only at the surface of the liquid. However, once the vapor pressure of the liquid exceeds atmospheric pressure, bubbles can start forming and the liquid boils.
This explains why the boiling point of a liquid is dependent on atmospheric pressure. The boiling point of the liquid does not change because a change in atmospheric temperature affects the vapor pressure; rather, it is the boiling point itself that depends on atmospheric temperature. Vapor pressure, as I mentioned before, is entirely independent of atmospheric pressure.
(as an aside, there is some pretty cool physics/chemistry involved in the formation of bubbles. As it turns out, the limiting step of bubble formation is whether the internal pressure of the bubbles can overcome the surface tension of the water [i.e. the intermolecular bonds that hold water molecules together]. Interestingly, surface tension forces are largest for infinitesimally small bubbles and decrease as the size of the bubble increases. Most of the time, bubbles need to be nucleated; bubbles need to form along a surface where a scratch can help disrupt the surface tension of water and allow a larger bubble to form. One consequence of this is the ability to create superheated solutions; heating water in a very smooth container will allow water to be above its boiling point because the vapor molecules will not yet have enough energy to overcome the surface tension of water and there are no sites for nucleation. Disrupting the surface tension of water, for example, by stirring the solution, nucleates bubbles, often rapidly and violently, causing an explosion of vapor. This is why you need to add boiling chips to any solution that you heat in laboratory glassware. Similar phenomena involving surface tension and nucleation are important in crystallization [and help lead to the fractal patterns you see in snowflakes])