What is the difference between a holdings company and a private equity firm?

I mean blackstone buys over companies and they are considered a private equity firm. But BerkshireHathaway also buys over companies but they are considered a holdings company.

Very simply put, a holding company is a co. which owns 51% or more in the equity of another co. (called its subsidiary). Holding companies are usually for the long haul, and can exist for decades (think Coca Cola).
Private equity firms invest in non-public companies and typically hold their investments with the intent of realizing a return within 3 to 7 years. Generally, investments are realized through an initial public offering, sale, merger, or recapitalization. Private equity groups tend to focus on more mature businesses, often contributing both equity and debt (or some hybrid) to the transaction. The firms were commonly called leveraged buyout firms (LBO) in the 1980s.

As you can see from the following definition of consolidation in the context of private equity firms, they can form holding companies to achieve their purpose. Thus, a private equity firm can be a holding co. The 2 are not mutually exclusive.

Consolidation : Also called a leveraged rollup, this is an investment strategy in which an LBO firm acquires a series of companies in the same or complementary fields, with the goal of becoming a dominant regional or nationwide player in that industry. In some cases, a holding company will be created, into which the various acquisitions will be folded. In other cases, an initial acquisitions may serve as the platform through which the other acquisitions will be made.

The following articles may help you understand this better.