They are bascially the same, the only difference is that "firm" is more formal and used with certain professions, e.g Law Firm

Firm:専門職または小さい会社 Company:一般的な「会社」というイメージ

A company is typically a corporation owned by a large group of shareholders, who individually have limited liabilities. A firm can refer to a partnership, like an architect or law firm, typically owned by fewer people than a corporation, which can have unlimited liabilities for the partners. A firm can refer to a corporation as well, but a company cannot refer to a partnership.