Productivity curves will show you how a company's productivity changes as variable inputs are increased. For instance, how much can you produce when you hire 4 workers? 5 workers? 6 workers?
Cost curves will show you how a company's cost changes as production is increased. How much will is cost to produce 10 units of output? 20 units? 30 units?
Productivity curves and cost curves are inversely related. This means that as marginal product increases, marginal cost diminishes and as marginal product diminishes, marginal cost increases. You can think about it like this - As we become more productive(i.e. hiring more workers), our costs per unit of output will diminish, encouraging us to keep producing. When our productivity begins to diminish(i.e. we've hired too many workers), our cost per unit of output begins to increase, encouraging us to stop producing.