Carve-Outs
Parent company sells some or all of the stock of a subsidiary to the public in an IPO
The carve-out may pay a portion of the IPO proceeds to its parent
Parent companies sometimes link subsidiary IPOs and Spin-Offs (two step spin)
Parent would typically sell less than 20% of subsidiary to the public and then distribute the balance of the stock to their shareholders in a tax-free distribution (Example: Bristol-Myers Squibb / Mead Johnson Nutrition)

Parent sells Equity in the New Firm to the Public (IPO) and creates a New Publicly Traded Entity.
A carve-out brings cash into the firm, whereas a pure spin-off does not.
Carve-outs disperse assets and ownership in the assets to non-owners of the original firm.
Carve-outs are often an intermediate step before a full spin-off.