One of the constituent elements of illegal fund-raising is to raise funds from unspecified people (the public) without legal procedures.
Therefore, there is a big difference between the two:
A. Private equity funds raise funds for specific people, not unspecified people (the public).
B. Private fund managers raise funds according to the Securities Investment Fund Law and other laws and regulations, and filing is only the latter procedure.
Conclusion:
The two are different categories and have nothing to do with illegal fund-raising. If private fund-raising is aimed at specific people, it is certainly not suspected of illegal fund-raising.