What is the difference between a Venture Capital firm as compared to a Venture Capital fund?

Generally speaking, a Venture Capital firm is a reference to one or more Venture Capital Funds managed by an overarching entity (or sometimes a group of entities). On the other hand, a Venture Capital fund is a specific pool of capital that the firm invests. It is a blind pool, meaning the LPs do not know what deals are going to be invested in, and it contains a diversified set of portfolio companies. Venture Capital firms typically aim to manage many such pools of capital as they grow and scale. 

For example, Sequoia Capital is one of the most high-profile Venture Capital firms in the world. However, it manages many, many Venture Capital funds under that brand.