I've seen many videos of lift-offs etc (plus I'm huge Star Trek fan), but this one, made me hold my breath for a good 45 seconds (thank you yoga!)
For the longer approaches, the rocket launches with an offset of several (tens of) minutes of that instantaneous launch window (that mostly means you have a launch window of several minutes, i.e. short delays do not abort your launch), so the spacecraft comes up significantly „in front of“ or „behind“ the ISS. As with the fast approach, the spacecraft will still not launch into the same orbit (i.e. mostly lower and more elliptical than the ISS orbit). But this is more important for the slow approach: Due to orbit dynamics, the difference in orbit will make you „catch“ up with the other space vehicle over time – this is what is called „phasing“. When you’ve aligned the phases, you raise/align the spacecraft’s orbit to the ISS and are thus very close and in final approach.
If your phase is not well aligned at orbit insertion (i.e. just after the launch), you need to do more and thus longer phasing. I.e. the bottom line is that you need a high precision launch, precise navigation sensors & enough computational power on the rocket and spacecraft to be able to do a fast approach in 2-3 orbits, plus the ISS orbit must align very well with your launch site at launch time. It is simply more complex and thus more risky as the slow approach, but then again you are faster if everything works out perfectly. The safer bet is the slow approach, though, as it gives you more margin and more time to fix any issues that arise.