Th cells are antigen specific. There may literally be just 10 or 20 cells able to target a particular antigen on a microorganism, so the first response to activation by antigen presenting cells is to multiply in numbers. For as long as the T cells receive antigenic stimulation they will continue to proliferate. These memory cells don't get involved in the attack on pathogens rather, they act as the filing system to record that the immune system has encountered this particular antigen. The memory cells will act as the cell pool for future proliferation if even more T cells are required.
T cells are what vaccines ultimately activate and produce.
The presentation of vaccine antigens to the immune system via antigen presenting cells induces those Th cells that are specific for the particular vaccine antigens to proliferate. Most of the T lymphocytes will be effector cells which will quickly set to work to remove the vaccine antigens. Some of the cells will be memory cells and they now keep the code for the antigens in the vaccine. The memory cells respond far better to antigenic stimulation than virgin lymphocytes. They proliferate more quickly, produce more daughter cells and are more vehement in their attack on the pathogen.