Step-down converter modes
As you say, in continuous mode, the inductor current never stops, while in discontinuous mode, there is a period of time each pulse when there is no inductor current.

To get into why each one is used, advantages and disadvantages of each, would take a whole book.  Most likely, there are actually books written on this.  If you really want to learn this, then you should go read one.

However, briefly, some of the issues of continuous versus discontinuous operation are:

The ripple can be made lower in continuous mode.  With high enough switching speed, the inductor current only goes up and down a little each pulse.  This delivers more steady current to the output.

The peak inductor current is less in continuous mode for the same load current.  This means a smaller inductor can be used.  The inductor is often the largest and most expensive component in a switching power supply.

Continuous mode is difficult under low load.  For any fixed switching frequency, inductor, and input and output voltages, there is a load current below which continuous mode can not be maintained.  This means any continuous mode system either has to specify a minimum load current, or be able to deal with discontinuous mode for some load conditions.  This is not always as simple as it sounds because the open loop transfer characteristics change between the two modes, and that may require the compensation scheme to change.

Continuous mode is a bit trickier to make stable, at least when pushing the limits.  This is grossly oversimplified, but consider reacting to a sudden decrease in load.  Continuous mode always leaves energy in the inductor at the end of each pulse.  It therefore can't stop dumping current to the output as quickly as discontinuous mode, which by definition stops dumping current at the end of each pulse.  This implies a slower open loop response, which in general means its harder to keep stable, all else equal.

In discontinuous mode, the diode is not conducting when the switch is first turned on.  At low voltages, Schottky diodes can be used, which have very fast reverse recovery time.  However, not having to deal with reverse recovery at high voltages can be useful.  Consider that when the diode and switch are both on, they create a short across the input.  That is not only bad for efficiency, but can abuse the switch and/or the diode.

Really simple control schemes, like pulse on demand, aren't so simple anymore in continuous mode.  Pulse on demand results in more ripple, but is inherently stable and very simple to implement.  It's simpler to design something that stays nicely predictable and that you know won't saturate the inductor with discontinuous mode.



In practice, when I'm using a off the shelf chip to implement a switching power supply, I don't really care what mode it uses or when it switches between them.  What you care about usually are things like ripple voltage, load and input regulation, and the like.  How that's accomplished is (largely) irrelevant when plunking a chip on a board and following the datasheet recommendations for inductor and the like.

Sometimes I implement a switching power supply by having a microcontroller produce the pulses.  That's often cheaper than a switcher chip, especially if the micro is already there for other reasons and it has a spare PWM output with maybe a shutdown input or comparator input, and ripple and the last possible bit of efficiency aren't that important.  In such cases, a pulse on demand system or something that just shuts down the PWM when the output is above the regulation threshold is simple and good enough.  If the system runs in discontinuous mode, it is easier to analyze and know the inductor won't be saturated.

You mentioned a flyback converter in your question.  Flyback converters are generally discontinuous, else you're just wasting power by keeping a bias level of DC always running thru the primary.