HIV virus inserts itself into the DNA of long lived cells in your organism, and it stays around as long as those cells stay around, which is (unless u have blood cancer and your doctors decide to kill those cells) your entire life.
That constant low level stream of HIV viruses in your body can cause the virus to return at any moment you stop the therapy. So you have to take daily pills for the rest of your life to stop that stream from exponential growth and killing all of your immune system.
The last part is what is referred to as AIDS: your immune system is so weak that even the simplest pathogens can kill you, which they usually do once you reach AIDS stage.
There are a couple of viruses that can lay dormant in you and cause problems later on. Many of them insert themselves into the DNA of long living cells.
Without therapy, HIV will progress to AIDS in a couple of years in basically 99.9% of patients. Some very rare cases are known where HIV patients survive without getting AIDS and therapy, via broadly neutralizing antibodies. But again, less than a tenth of a percent. Everyone else would die without a therapy.
Being HIV positive isn't a death sentence, if you take your meds (and started taking them early after HIV infection) you can live for as long as someone who isn't HIV positive. In fact studies show that the effect of regularly going to a doctor and getting your blood checked actually improves your life chances. HIV positive people can do 99% of the things normal people can, although they are not allowed to donate blood (for good reason).
Having AIDS though is a major impairment and you are on your way to death at that stage (1-3 years to live). So it makes sense to distinguish the two.