I come across such distinctions in quite a few places such as in this Wikipedia article, Voiceless glottal fricative, where it states: "The voiceless glottal fricative, sometimes called voiceless glottal transition, and sometimes called the aspirate,1 is a type of sound used in some spoken languages that patterns like a fricative or approximant consonant phonologically, but often lacks the usual phonetic characteristics of a consonant". Of course it is clear that a phonetic constant would have a "complete or partial closure of the vocal tract". So when would a sound be a phonological consonant while its status as a phonetic consonant is doubtful?
The notion of a "phonetic consonant" as distinct from a "phonological consonant" is rather anomalous. "Consonant" is a kind of segment, as is "vowel", and all three terms (including "segment") are phonological concepts. Phoneticians don't define consonants, or vowels, they take those phonological units for granted and ask phonetic questions about consonants, or vowels. So there's no such thing as a "phonetic consonant", distinct from a "phonological consonant".
The so-called "usual phonetic characteristics of a consonant" emerge from throwing all consonants into a bag and seeing what things are true of at least half of them, thus it's a "usual" phonetic characteristic of a consonant that it is oral, but there are nasal consonants. By a slim margin, consonants are "usually" produced with continuous airflow through the vocal tract, and by a wider margin, witout significant pressure buildup in the vocal tract. The issue with h and a few other consonants is that they have relatively little constriction in the vocal tract, and may be terms "approximant". It's not that the status of [ɹ j w h ʔ ʕ etc] is in doubt, phonetically, it's that they have properties not true of most consonants, just as [m n ŋ] have properties not true of most consonants. They are more like vowels – similarly, [ĩ ã õ] are "more like [m n ŋ]" that [i a o] are. In the case of "h" (also ʔ), because there is no supraglottal constriction, that consonant contributes nothing to the resonant frequency of the vocal tract, which then has lead some people to talk of h as being a voiceless / breathy version of the following vowel (when there is a following vowel).
In phonology, there has been some debate over the so-called major class features that define consonants, vowels, glides and so on leaving out place of articulation. The standard SPE solution is three features, [consonantal], [sonorant] and [syllabic], which allow you to define all of the "kinds" of segments including syllabic consonants like [m̩, l̩]. However, the featural status of syllabic fell into serious disrepute in the 1980s in connection with the segmental-length debate and the connection between "being syllabic" and "contributing to length". The feature [consonantal] also came to be questioned on the grounds that it was not really well motivated in phonology, and seemed to be replaceable with a subtly different classification of place of articulation which made sense at the time. In general, phonologists have not been happy with the theory of major class features because they don't seem to be well-motivated as compared to "voiced" and "nasal".