Normal symptoms of a honey bee sting include swelling, redness, and pain at the site of the sting. Systemic symptoms, or reactions that occur where you didn't get stung, are of greater concern. A drop in blood pressure is the most dangerous symptom, because in this case blood isn't being pumped into your brain, which can lead to death.
No matter the symptoms, if you are in doubt of the severity of the problem, use the Epipen! It results in a feeling like you drank too much coffee, but these symptoms pass in about 20 minutes and are preferable to a severe allergic reaction. About 25-30% of people need a second dose, so it is a good idea to keep two around.
Dr. Copenhauer also suggested that a "feeling of dread" is a common indicator that the sting is very severe. In this case, it is better to administer epinephrine than to wait to see how the symptoms play out. Interestingly, the "knock-off" brands are not as effective. Additionally, even if the pen has passed its expiration date, it is better to use it than to wait. Some ibuprofen might also help as an anti-inflammatory in case epinephrine is not available.