The only difference between 1st and 2nd is precedence in the absence of the bishop.
The two counselors take turns with the bishop at the pulpit of services, and probably their biggest anxiety is whether the assigned speakers show up...more than once I had to give a talk in the spot because the speaker was a no-show. They collect/deposit tithing and have check-signing authority They do temple recommend interviews They discuss who to assign to various ward positions in the ward They teach classes sometimes They do church welfare errands, like dropping of checks
In our case, he also likes to drop in unannounced and check out whats going on at our house. He usually brings 2 or 3 or his small children with him. He talks about fishing with my husband, but never says a word to me.
The two counselors also split all the auxiliaries (except the Relief Society, which the bishop handles) in terms of supervision and extending callings. One counselor would be responsible for extending callings to Primary teachers and the other to Sunday School teachers, for example.
The 1st counselor also oversees the Teacher's quorum and the Mia Maid class. (the 14 & 15 year old boys and girls) The 2nd counselor oversees the Deacon's quorum and the Beehive class (the 12 & 13 year old boys and girls).
Oh, and be the bishop's lap dog and do anything else he asks you to do.
(I've been 2nd counselor to two different bishops and I know whereof I speak.)