Let me put it without much math. Consider multiplication: As and when you get a number, you multiply it by one more number and pass it off as output. A multiplier does not remember stuff. Its like a small child. It forgets all the things you did in the past and responds to the current input. If you smile, it smiles back, so forth.
But, a convolution is different. It remembers your past actions. Based on that and your present doing, it will decide what to do. If you smile at it and you did real-bad before, it would not smile back at you!
Another analogy, I can give is of a Resistor and Capacitor. V-I relationships for resistor is just multiplication. But, capacitor remembers past actions in a sense that it stores charge (which is due to all past inputs) and hence, when you work with them, you need to convolve!