I am having a little trouble understanding the actual difference between these sets of sounds. What would be the difference in pronunciation between /hau nau braun kau/ and /haw naw brawn kaw/?
There is no difference that is intrinsic to these transcriptions. IPA letter do not represent exact pronunciations of a Platonic linguistic metalanguage, they snad for ranges of pronunciations in actual languages -- but each pronunciation is in a specific language. Languages can differ substantially in how <hau> is pronounced. The meaningful difference is not in the pronounciation of the string of letters, it is in the phonological analysis. The letters <w,j> are conventionally consonants and <u,i> are vowels, so writing <hau> implies that there are two vowels, and writing <haw> implies that the word ends in a consonant. There aren't particularly strong arguments for one analysis over the other in English. In some languages, there are good tests that distinguish between CVV and CVC, but the high postvocalic vocoid of English has been analyzed both ways.
It's not an answerable question. There is no general understanding among phonologists about such notational differences -- you have to know the theory or the intent behind the use of the notation.
You use square brackets in part of your question, which are generally used for phonetic transcription, and slashes, often reserved for phonemes, in another part of the question. That's confusing. I'm going to assume you mean to be asking about phonetics.
A phonologist might have noticed a difference in place of articulation between syllable onset [i], as in "yacht", which is similar to the vowel [i], and syllable offset [i], as in "tie", which is more like a barred-i. A reasonable use to put the alphabetic distinction between vowel and consonant would be to use [i] for the offset sound in "tie" and [j] for the onset sound in "yacht".
C.-J. Bailey (in Southern States English) once gave a minimal pair for the onset and offset sounds in "Maya" (as in "Angelou") versus "Maya" the mesoamerican indians, which following the above could be written [maia] versus [maja]. However, Bailey's idea was to distinguish the two by where the syllable boundary comes: [maj.a] versus [ma.ja].
Another idea, due to Ilse Lehiste and probably other phoneticians, is to classify such sounds as vowels when they have a prolonged steady state, during which the articulation remains constant; otherwise, as glides.
Yet another plausible use of the notational difference [i/j] is according to syllabicity. In casual speech, intervocalic syllable offset alveolar stops can be flapped in some English dialects, and even lost entirely in faster speech, giving rise to two syllable pronunciations for a word like "toddy", [tai], which is still distinct from single syllable "tie", [taj].