The time of year you travel to Antarctica does make a difference. However, deciding what time is "best" depends on what you are personally desiring to get out of your adventure. Particular wildlife sightings - mating season - new penguin hatchlings - specific weather - icescapes - mud - penguin poo smell at its worst/best...?
November: The month of November (late Austral spring and early Austral summer) undoubtedly offers the most adventurous time to visit the Peninsula. This is the time to see Antarctica in its most undisturbed form. As the season goes on, the landing areas become impacted and muddy. Some operators tend to discourage November voyages as temperatures tend to be colder and polar ice is still breaking up, so access to some areas may be limited. However, the cold temperatures also offer the most impressive icescapes, pristine snow and breathtaking scenery. Wildlife -- particularly whales (however, I have personally been in November and we had five humpback sightings) -- is typically not as plentiful as later in the season. Although it is a good time to observe the penguins mating. By the end of November you'll see lots of nests full of eggs!
December to Early February:
December and January are the months with the most sunlight (up to 20 hours a day), and daily temperatures are at their warmest. In January, temperatures at the Peninsula average 34 Fahrenheit. Penguins begin hatching and wildlife, especially penguin chicks, is most plentiful during this time of year.
Mid-February and March:
Late summer in Antarctica, February and early March, is the best time to spot whales. However, you are taking the risk that other wildlife may already be gone out to sea. By March, the chicks are quite large, and they start to fledge. Weather wise, temperatures are beginning to drop as the season starts to change. Daily highs average roughly 29 Fahrenheit on the Peninsula. There tend to be less vessels operating at this time, which means you won't have to compete with other ships for landings. This time of year you are also likely to have better access to areas further south as polar ice melts. You will rarely walk on snow during this time; expect rocky and muddy landings.