Did one immediately get more power in some way? Did one all of a sudden have more responsibility? Any particular perks? How, specifically, did a person’s life change the day after they joined the communist party?
It very much depends on the time you're asking about. In 1918 it would have been an open declaration of your political views. In 1925 it could have been pure fanaticism or a desire to make a career, and it could have given additional benefits in the form of groceries, etc., which the old Bolsheviks didn't like very much. "Stalin turns the Party ticket into a food stamp!" In 1941, in the occupied territories or in the army, having a Party card guaranteed you, at best, a firing squad on the spot if you fell in with the Germans.
There is a legend I read three times about different people which also applies to this time. "My grandfather always refused to join the party, but he applied for joining in 1941-1943, when his military unit was encircled and he saw their political instructor burying his party card". I met this story in at least three different memoirs of different people, so you could say it was some variant of the urban legend. Well, in general, it was really dangerous to join the party back then. I know a story that after the war, somewhere in the 1960s, a man was fired from a high position of Deputy Minister and retired, when the story goes that there was a danger of being taken prisoner during the war and he buried his Party card.
If you're asking about the late Soviet period, joining the Party was required for leadership positions and careers. For example, one could have a career in science without party membership, but one could not become director of an institute or hospital without it.