They are identical cartridges.
Many cartridges have multiple names.
The 9mm Luger is also called the 9X19mm and 9mm Parabellum.
After World War I, when returning Doughboys brought back Lugers as war souvenirs, American factories began producing ammo in greater numbrers. It had been made before World War I by some American ammunition companies but it was practically a special order item and production was low.
In those early days, at least one catalog referred to it as the 38 Luger!
Today, it's most often called the 9mm, 9mm Luger, 9X19, 9mm Parabellum or 9mm Para.
The Sporting Arms and Ammunition Manufacturer's Institute (SAAMI) establishes the maximum average pressure of ammunition manufactured in America.
SAAMI recommends a standard of 35,000 pounds per square inch (psi) for regular 9mm ammunition.
SAAMI also sets a maximum average pressure for +P 9mm Luger ammunition at 38,500 psi.
Such +P loads will accelerate wear on your pistol if fired frequently.
Some pistols may not be strong enough to handle +P loads, or may require a stronger slide spring to compensate for the increased velocity of the slide; check with your manufacturer whether your pistol is rated safe to fire with +P ammunition.
Also beware of 9mm Luger military ammunition loaded to North Atlantic Treaty Organization (NATO) pressure standards, which are higher than American commercial ammunition.
This ammo will have the NATO symbol of a small circle with crosshairs in it, stamped into the base of the case.
Sources vary on the maximum average pressure for various NATO 9mm cartridges; they range from 37,000 psi up to 42,000 psi.
This is not surprising, considering that various European countries that belong to NATO are making ammunition for their own troops.
Many claim that this "hot" NATO ammo is made for submachine guns, which can handle higher pressures than pistols, but my research indicates that ammunition assembled solely for submachine guns is comparatively rare.
This makes sense when you think about it.
Troops in combat will almost certainly not take the time to discern between "pistol" and "submachine gun" ammunition, so why create ammunition that might blow up their pistols?
Why create two standards of ammunition to track and distribute, requiring great care?
A soldier who damages his sidearm by using submachine gun ammunition in it not only endangers himself, but may be left totally defenseless.
True, some 9mm ammunition has been assembled by various militaries to be used only in submachine guns, but I don't believe it's as common as some believe.
From what I've been able to learn, most countries make their 9mm ammunition suitable for both submachine gun and pistols, though it may be near the upper limit for pistol use.
Commercial American 9mm ammunition (Winchester, Remington, Federal, etc.) that is NOT +P is not as high-pressured as most European commercial ammunition.
Very good ammunition is made by Fiocchi of Italy, Sellier & Bellot of the Czech Republic and DWM of Germany but it's obviously loaded to slightly higher pressures than the SAAMI standard of 35,000 psi.
While the European ammunition may not equal the +P loads of 38,500 psi it is probably near it. Or it may equal or exceeed it. Lacking a laboratory to determine the pressure, I can't say -- nor can anyone else who lacks such sophisticated equipment.
Case condition, velocity, flash and report are not good indicators of pressure.
Until +P ammunition was introduced by SAAMI in 1974, American 9mm Luger had always been loaded to lower pressures than European ammo. This was in deference to the occasional worn out or comparatively weak 9mm pistols that returning American soldiers brought home, or were imported to America.
Interestingly, many German Luger pistols will not function well with American ammunition because American ammunition is loaded to a pressure level below what they were designed to use.
For this reason, the Luger pistol earned a reputation for unreliability that is somewhat unwarranted.
Right after World War II, the American Rifleman magazine made note of this problem and suggested that owners of Lugers clip a few coils off the recoil spring, to make the spring a little weaker and the Luger more reliable with American ammo.
The American Rifleman also noted that, once doing this, it would NOT be a good idea to use European ammunition in the Luger again; such ammo would now be too powerful and batter the gun.
So, should you buy or inherit an old German Luger pistol in the future, you may wish to check the length of the recoil spring. If it's original length, use European ammo for best reliability.
I would NOT suggest using +P 9mm ammo in a Luger; it may be too powerful and batter the gun.
So there you have it, a little background that will help you purchase the proper 9mm Luger ammo for your gun.
I would suggest you purchase commercial, regular-pressure ammo from Winchester, Speer, Remington, Federal or other reputable American firms.
For practice, 115 or 124 gr. full metal jacket works fine, though many indoor ranges will not allow it. If you plan to shoot at an indoor range, find out what they allow then purchase your practice ammo based on that.
For self defense, you'll want a jacketed hollowpoint or softpoint. Few shooters can agree on what's best for defense, and millions of herds of dead horses have been flogged to dust in debate, so I won't get into it here.
But I will say this: Only hits count. The finest bullet in the world is useless if you miss your aggressor. The latest Super-Dooper Turbo ZX 1000 DragonSlayer bullet can never be a substitute for proper bullet placement.
Avoid off-brand, reloaded ammo. While it may not be dangerous, it is often hurriedly assembled and quality suffers, especially if it's loaded with lead bullets.
Getting back to your original question, you will find that many calibers have various names. In Europe, the 7.62X51R is what we Americans call the .30-30 Winchester, and the European 7.62 X 63mm is what we call the .30-06.
Good luck with that 9mm Luger ... 9X19mm ... 9mm Parabellum ... whatever ...