Power
You don't gain any John Constantine-like psychic abilities, but you do become powerfully informed. Being a private detective gives you the ability to delve into other people's secrets. There are many different opportunities for private detectives to gain access to people's dirty laundry.
Legal investigators are the detectives of the lawyer world. They help verify facts in civil lawsuits, prepare criminal defenses, and serve legal documents. You get to say, "You just got served," without talking about street dancing. However, you should walk away quickly after saying this. People aren't going to be too happy when you throw it in their face like that.
For the savvy hackers, computer forensic investigation is an option. You'll recover, analyze, and present evidence from computers that are then used for investigations. These computer geniuses could even find that deleted email you wrote to your boss five years ago.
Other empowered private eyes get the power of a lot of cash behind their investigations. Financial investigators are usually hired by business bigwigs to create confidential financial profiles of companies and individuals that are part of large transactions.
Meanwhile, corporate investigators help companies by conducting external and internal investigations. They bust people who are abusing expense accounts, expose drug use in the workplace, or investigate fraudulent billing practices.
Whatever your specific field of investigation, you're going to have access to a lot of private and confidential information. Use it wisely.