In simplest terms, as you climb, the air pressure around you decreases and your watch subsequently registers an increase in elevation.
However, thunderstorms are often associated with a low-pressure weather system moving in, which will also register on your barometer. This can erroneously be interpreted as an increase in elevation. Similarly, high pressure (clear weather) systems moving in can make your watch register a decrease in elevation.
Some modern watches have accelerometers in them that can sense when you are moving. Algorithms then determine whether a change in pressure is likely due to your climbing (moving, and in an upward trending motion) or due to weather (no movement). These calculations can increase accuracy significantly, but you'll still need to calibrate on a regular basis.
Some GPS watches like Garmin's Fenix 3 (which has an acceleromter) also can auto-calibrate elevation based on GPS information -- though GPS elevation is notoriously bad.
More accurately, the Fenix 3 can also use your smartphone connection to look up your elevation based on USGS survey data of your current location (given by the GPS). This is not automated, but can allow you to more precisely calibrate the watch as often as you feel necessary.
Without super-sensitive tech to register these pressure changes, you'll be hard-pressed to track this on your own in any meaningful way.
Finally, temperature does have some (albeit slight) effect on mercury-based barometers as the the density of the liquid changes causing variation in the readings (low pressure = more volume). Modern non-mercury barometers are not influenced by this to any significant amount.