By 'empirical', Popper means 'a posteriori'. By 'logical', Popper means 'a priori'. The distinction between a priori and a posteriori is a distinction of kinds of truth/falsity rather than kinds of propositions. On the other hand, the analytic-synthetic distinction is a Kantian distinction about the subject-predicate relations of propositions. Examples of all of these will depend on the particular philosophy.
First, I'll give examples distinguishing a posteriori (i.e. empirical) from a priori (i.e. "logical"):
- If someone tells you that they're happy then it is true a posteriori that someone told you that they're happy.
- If every natural number is the sum of its preceding whole number and one then it is true a priori that 5=4+1.
Next, is the analytic-synthetic distinction. Analytic statements are those in which the subject's concept strictly implies the predicate's concept, and synthetic statements are those in which the subject's concept does not imply the predicate's concept. I'll give examples distinguishing analytic from synthetic:
- "The eukaryotic cells are animal" is a synthetic statement, because many non-animal species (e.g. the entire plant & fungi kingdoms) have eukaryotic cells such that the subject "eukaryotic cells" does not imply the predicate "are animal".
- "Three is a number" is an analytic statement because the subject "three" strictly implies the predicate "number".
Popper is pointing out that, when these are not properly distinguished by scientists, category mistakes like "it was before the beginning of time" or "something came from nothing" can be made. "It was before the beginning of time" is false a priori because "before" has the extension "is at a time that is followed by" such that "before" is in the category "time" and, thus, it is a contradiction for there to be a time before time if the extension of time includes every referent for "before" and "after". "Something came from nothing" is defective in that identifying any referent for "nothing" falls into a paradox of self-reference and, moreover, such a statement usually is a vague way of saying, "Something presently known came from nothing presently known", which is a categorically different idea from "something came from nothing".