The above answers are correct. Perhaps I can be a bit more concise and specific, and cover some ambiguous meanings.
The two words are commonly used interchangeably.
"Caliber" is properly the diameter of the bullet, in hundredths of an inch. In fact, caliber means (in one sense) a specific measurement in decimal fractions of an inch (which is a standardization in the Imperial measurement system).
Or sometimes the diameter of the bore of the arm. To further add to the confusion, the United States manufacturers tend to refer to groove diameter as the proper measurement while Great Britain manufacturers (there is not governmental decree of which I am aware to mandate either) measure land diameter. Which is why a "6.5mm" rifle is '.256 inch' in Great Britain and '.264 inch' in the U. S.
"Caliber" is also used commonly to designate the size and shape of the entire 'package' of a unit of ammunition. When asked "What caliber are you shooting?" One commonly answers ".270 Whiffinpoof" or such, usually without pronouncing the decimal point.
"Cartridge" is properly the entirety of a loaded unit of ammunition, ready to be inserted and used in a firearm. As you remarked, there are many "30 caliber' rifles; they all have the same bore diameter. In the U. S. they all share the groove diameter of .308 inches, but not in all the world; notice the bore diameter of the Russian 7.62x54Rmm cartridge. In 'Mother Russia' this round is considered a "three line" (meaning .30 caliber in the Russian language of 1891) but by our standards the bullet and bore diameter is .310" rather than .308".
On a similar note, the term 'round' has an equally ambiguous meaning. The late Jeff Cooper wrote to the effect of a 'round' being ammunition sufficient for all 'guns' to be fired once. This was a military phrase and concept. In the dictionary, a 'round' is sufficient material for one shot. In the case of a muzzle loader, an amount of powder, one projectile (or a load of shot), wadding as needed, a percussion cap and I'm not sure about flint if so required. The current concept is a single unit of ammunition, ready to be loaded and fired.
One can also speak of a round in the sense of a cartridge; the .30-06 round.
Much like 'magazine' and 'clip' are (wrongly and) interchangeably used, the English language is at the mercy of the sloppiest speaker. One is encouraged to listen to the context of the statement and use of any single word.