Firstly, legal ownership of the IP. The certificate of registration will be the proof that you are the owner of the IP, be it a brand, patent, copyright or product design. There is no better proof than this.
Further, registered IP owner is the only one with the rights to use, manufacture, sell, license and reproduce the IP, respectively.
Registered IP owners may also sue the infringer for damages and/or Court injunction to prohibit further infringement.
In addition to civil actions, infringement of IP is also a criminal offence in certain jurisdictions which upon conviction will be liable to fine and/or imprisonment.