Is there any difference between investing in a mutual fund and in an initial public offering of a company?

Yes, there is a difference. IPOs of companies may open at lower or higher price than the issue price depending on market sentiment and perception of investors. However, in the case of mutual funds, the par value of the units may not rise or fall immediately after allotment. A mutual fund scheme takes some time to make investment in securities. NAV of the scheme depends on the value of securities in which the funds have been deployed.