You've understood the two related meanings of count - to enumerate or to measure, as opposed to to matter or to be important. A gloss of the statement might be as follows:
Many of the things you can count, don't count. [Many of the things that can be enumerated are unimportant.]
Many of the things you can't count really count. [Many of the things that can't be enumerated are important.]
To see the difference between things you can count and things that count, it's important to note that native speakers will be unlikely to parse this quote with the reverse order. If they start to do so, getting the things that are important from the things you can count, a conflict will be reached as soon as don't count is encountered.
Things [that] don't count can pretty much only refer to things that are unimportant (see sense 3: we say it counts or it doesn't count). It is very much a stretch to imagine that don't count means the things don't have the ability to count (themselves?), or that they cannot be enumerated, because another way of saying that would seem much more familiar (things that are uncountable, for example, or the things you can't count we just read before this).
Similarly, can't count is probably the more difficult part of the next phrase to parse. It may be confused with the idea of counting on, meaning relying on someone or something, and one's mind may be looking for the word on; this could leave you thinking for a moment that things you can't count on, perhaps unimportant things, was intended. However, because the next part, things [that] really count unambiguously refers to things that count in the sense that they are important, again, any misunderstanding must be reconciled. It makes much more sense for things you can't count to refer to intangibles, things that can't be enumerated or measured, and really count to mean we're saying they're important. *
For more examples, have a look at really count and can count in COCA. Really count is an example where count is overwhelmingly used to refer to whether something matters or is important; can count most often occurs in references to actual counting or the idea of whether something is measurable:
the only opinions that really count are those of the couple in question
And that time at the Taos Inn doesn't really count, since she sent me away
bald eagles, blue herons, osprey, and more alligators than we can count
on the fingers of one hand I can count the people likely to be calling me at this or any hour
While there's some overlap and indeed we could say in many examples that the things that count (matter) can be counted (enumerated), such as opinions, the two senses cannot be swapped.
(The only way I can even think to use really count in the sense of enumeration is to say something like, Wow, look at your baby brother - so big already! He can really count!)