What is the advantage of a "compact crankset" vs changing a chainring? It seems that I'm old (my last good bike had a 5-speed freewheel and shifters mounted on down tube braze-ons - they were indexed, though!), and I've been out of cycling for quite some time.

I've seen several references to a "compact crankset" and questions like this one asking about replacing the whole crankset just to change gearing. Back in my day (get off my lawn you meddlin' kids!) I would have (and, occasionally did) simply changed a single chain ring to one with a more or fewer teeth to change the gearing. 

Why would one now change the complete crankset instead of buying a single new chain ring? Does it bring specific advantages to the table, or are replacement/different size chain rings simply no longer manufactured? (The cynic in me says that means more profit for the manufacturers.)
Getting though all the hype and testosterone - Compact cranks allow a smaller small chain ring as they have a smaller BCD. Modern tech means you can have a wider gap between chain rings, giving larger spread.

This means modern compact cranks with wide range 11 or 12 speed cassettes can get the same gear range and stepping as a triple 7 or 8 speed with just a double. 

Where you would switch to a compact is if you have decided you need a smaller chairing than will fit on a standard crank. However, you may still need a new Front derailleur and possibly a shifter.