Which is more important in a manager, efficiency or effectiveness? Is one truly more important than the other?
I would argue that neither of this is more important than the other. A manager who cannot make a firm's processes both efficient and effective will not survive.
Effectiveness is important because it has to do with how well a process does what it is supposed to do. If a firm is supposed to be making t-shirts, it needs to have a process that actually makes good t-shirts. Otherwise, the firm will not prosper.
Efficiency has to do with how many resources (particularly time and money) are used in making the product. If the t-shirt company manager sets up a process that makes good t-shirts, but which spends too much money in doing it, what good is that manager. His or her firm will be making a product but will be unable to sell it because of its high price.
A manager must be both effective and efficient. Nothing else is going to allow the firm to prosper.