For STAT 636 vs 656: they look superficially the same, especially when looking at the syllabus. However, the syllabus can be a poor representation of actual course content.

There are a number of major differences. In my opinion, a course can be, in increasing order of abstractness: applied, methodological, or theoretical.

• 636 is a "methodology" course where 656 is a "applied" course. A methodology course means that instead of intensively using the methods, we talk more about why they might work, when you would expect them to work (and when you wouldn't), and what motivates their use. The assumed understanding of probability and basic statistics is noticeably higher. As well, I use linear algebra and calculus more intensively in 636. I've had students successfully take the course who were weak in these areas, but you have to be willing to get up to speed real time.
• About a third of 636 discusses multivariate statistical topics that pertain to statistical learning and are not included in 656 at all
• Likewise, a good amount of 656 is spent talking about computing/software topics (currently, using R).
• For 656, I spend much more time discussing the (important) details of actually using these methods in practice.

I have had numerous students take both classes, typically in 636 -> 656 order (though the other order makes just as much sense, to me). These students have commented positively on the experience as they felt they came out of the sequence with a deep yet practical understanding of the topics. If I had to make a broad statement, I would say that if you took 636 and felt really comfortable with everything we discussed, 656 might not be too helpful. Likewise, if you took 656 and you found it even slightly too "technical", 636 probably isn't for you.