What's the difference between the two circuits? Separate breakers?

Yes - separate breakers - connected as per image.

Are these two circuits on opposite phases? If so, then you will have 240 volts between the two hots. What was the purpose of this wiring scheme?

Not quite sure, but I can tell you the black and red are both hot and each are 120V so correct - 240V between the 2. I think this was the predecessor to GFCI. I saw this video for more info. https://youtu.be/rapPtMscrI0 But it didn't tell me whether I could just tap off of one half of this split circuit ... (not sure if the white would balance?)

I just watched the youtube and I don't think the split wiring was a predecessor to GFCI, although it came earlier. I have used it for the reason the presenter said, to reduce the current in the neutral if two devices are plugged in to the opposite phases. If you don't need the second circuit, then just cap it off (or better, disconnect it at the breaker box). And yes, you can certainly install a GFCI receptacle and wire it to the circuit you did not disconnect. If you extend the GFCI protection to downstream receptacles, just connect a cable going to them to the other two terminals on the GFCI receptacle. Of course, you ignore the capped off the wire.

I'm guessing you are in Canada? Before 2005 in Canada, all receptacles serving kitchen countertops had to be the 15 amp split circuit type, and they had to be dedicated circuits; you couldn't put anything else on the same circuit.

I am not a code expert, but I know that in 2005 code was changed to mandate 20 amp GFCI outlets within 6 ft of the sink. I don't know if code has since been modified to extend the GFCI requirement to the whole kitchen nor if 15 amp split circuit receptacles are still allowed, but I am pretty sure that you can't add a single 15 amp receptacle to this existing circuit.

There used to be Canadian electrical code experts on this forum but I think they are long gone.

It may be a code violation to extend a non-conforming circuit, but it seems it would be entirely safe to do.

I think the purpose of the split receptacles is so that you can have two big draw devices plugged into a single receptacle and they wouldn't trip the breaker if they were run at the same time. For instance, a toaster and a kettle. Since they'd be on different circuits, it wouldn't be an issue.

Or if they weren't at the same receptacle, you'd at least know that the tops of every receptacle is one circuit and the bottoms are another. So you'd plug the toaster in the top receptacle on one side of the kitchen and the kettle into the bottom of another receptacle.

At least that's my belief.

As seabornman said, from a safety perspective you can tap off of one circuit and extend it without an issue. Code-wise, they might frown on it since the whole circuit is not up to current day code. And you could install a GFCI at the new location. But better yet would be changing your breaker to a GFCI breaker so that all of the receptacles are protected. But that might entail moving some breakers around because I don't think there are tandem breakers that are GFCI. You'd probably have to take up the two full breaker spots and move those outside circuits to a different location.

I used to use the split receptacle on a single circuit, where one half was switched for a floor lamp and the other not.

Yep, that's another common use for split receptacles.

There's nothing really 'dreaded' about them. I guess for a novice, it might throw you for a loop because it requires a little more understanding about what's going on, but they can be really handy.

If you really don't like the split receptacles in your kitchen, you could rewire the receptacles so that they aren't split. For instance, you could make every other receptacle in the chain on alternating circuits instead. Just have the unused hot wire in the junction box pass through without connecting to the receptacle. That way, you could also install GFCI receptacles in each box if you'd like (nothing can be attached to the 'load' screws since it's a shared neutral circuit, but individual GFCI receptacles at each box is fine).