Primary cells are cheaper. But they are usable only once. The A, AA and AAA type of batteries or pen cells we buy for gadgets in the houses are primary cells.
These cells convert chemical energy into electrical energy due to travelling of ions in the electrolyte inside the cell. Once all the chemical is exhausted, the cell does not produce any electromotive force. It has to be discarded. The chemical reaction proceeds only one way.
In secondary cells the chemical reaction producing the movement of ions is reversible by an external electrical charging process. The cells can be recharged usually a few thousand times. They are much more expensive than the primary cells.
Primary cells can give higher EMF and power than the secondary cells. They are used in torch lights. Primary cells can generate more of the harmful waste as their lifespan is less. Zinc carbon cells are primary cells. Primary cells do not pass current if they are connected in the wrong way.
Secondary cells are made of Nickel Cadmium or lead acid. These are used in cars, inverters etc. For weakly charged cells, cell reversal and damage to cell can occur, if they are connected in the wrong way.