Criminal Justice is a broad title, similar to Nursing or Teaching, there are many areas to go into under that title. You say you want to be a private detective. You would then want to go into law enforcement and this means working the streets as a patrol officer first and then working your way up to Detective or Investigator. Once you retire, that's were private investigators or detectives come in. They have the experience as law enforcement.
The goal of criminology is to determine the root causes of criminal behavior and to develop effective ways for preventing it. So...Why do people commit crimes and how can we stop them from committing these crimes.
Criminal justice uses these findings to deter or prevent crimes from happening. So...now that we know why people commit crimes, lets try to stop the crimes from happening.
Criminology also tends to be more research focused whereas criminal justice tends to be more hands-on (police, corrections officers, etc.)
If you want to be a Detective, either course of study would be beneficial to you because understanding why people commit crimes can help you be a better detective and possibly help you solve crimes. But if you aren't really interested in the WHY then Criminal Justice may be more interesting to you in developing the skills necessary to prevent crime.
I would suggest contacting your local college and getting the email address of a criminal justice professor. They may be able to help guide you to the proper program based on your end goal. Good luck