Productivity is always important for a firm. But I suppose that you could say that it is especially important if the firm faces competition (as most firms do) and during an economic slowdown.
The reason for this is that productivity increases profit. Whenever productivity rises, a firm's average total costs go down and profits go up. Alternatively, when average total costs go down, a firm can lower their prices. This second one, especially, is important in competition and economic slowdown. In both cases, it is important to be able to lower prices without losing profit.