I have been looking into words like "pure" and "cube" and I sometimes get /pʲɔɹ/ and /kʲub/, but other times I get /pjɔɹ/ and /cjub/. I know some Slavic languages make the distinction between those two, and I would like to eventually be able to articulate that difference, as well as learn which IPA representation is more accurate.
There is no IPA principle that prefers superscript rather than regular j based on pronunciation alone. Both [pj] and [pʲ] identify the same phonetic fact, but differ in phonological analysis. In writing [pj], the claim is that there is a cluster of consonants, the second being a palatal glide. In writing [pʲ], the claim is that there is a palatalized consonant. You have to call on some phonological theory to decide which is more appropriate for the facts of a language.
If a sequence like "pj" can appear anywhere p can, including before other consonants, people are more likely to posit palatalization as a secondary articulation and call the thing [pʲ]. If it only appears before vowels, it's more likely to be treated as a cluster p+j. You might have stronger evidence for a cluster analysis if for example the language allows only two consecutive consonants and breaks up three-C clusters with an epenthetic vowel, and you find that /pjt/ becomes pjVt] but /pj/ remains unchanged. Contrarily, if you find that /p/ becomes "pj" after a front vowel (thus ip → ipj), it is more likely that people would treat that as secondary articulation, thus [ipʲ]. However, these kind of arguments are not particularly compelling on their own, and only hold relative to some concept of what a possible phonological rule is.
Some people, for example Steriade at different times, have tended to deny the distinction between a cluster and complex segment. Fujimura's CD model, in fact, rejects the distinction, in favor of complex articulations.