If you're apologetic, you're very sorry about something. You might offer an apologetic smile at another driver if you accidentally honk at her.
Someone who's apologetic is regretful or contrite, and usually openly expresses this feeling in some way. An apologetic note might say, "I am so sorry I picked all the flowers off your rose bush!" Apologetic was originally a noun, and its meaning in the 1600s was much closer to "justification" than "apology." It comes from a Greek root, apologetikos, "defensible." It didn't come to mean "regretfully acknowledging failure" until 1855.