Most of all, the two letters differ by one being considered a consonant and the other being considered a vowel. If that seems circular, that is because there is a certain circularity in the use of these letters.
Obviously, in the case of a single vowel sound between consonants or between a consonant and the beginning or end of a word, only i can be used.
By spelling tradition, a diphthong that ends in the [i̯] or [j] sound will usually be represented with i. On the other hand, a diphthong that begins with that type of sounds will usually be written with j. A word such as Leier is typically considered to consist of the syllables Lei and er while a word such as Koje will be considered to consist of the syllables Ko and je, meaning the glide is to the i in the former or from the j in the latter. However, this distinction is not fully strict. The name Hoyer from a local councillor (with a y, just to complicate matters) can sound as much as Ho-yer as it can Hoy-er.
In some cases, one encounters monosyllabic consonant-i-vowel or consonant-j-vowel sequences where the decision between the two seems really arbitrary. And essentially, it is. The only thing that might come to mind is that a native German may be tempted to read the i as its own syllable, especially in slow or dictating speech. However, there are also cases such as Diadem which is clearly a three-syllable word.
In the case of Sowjet, given the spelling it is typically analysed as Sow-jet. If it were spelt *Sowiet, a case could be made for analysing it So-wi-et or So-wiet which usually doesn’t happen – although those more familiar with Russian might still tend towards a So-wjet-style pronunciation. Am I getting nowhere? Correct because we cannot get anywhere.
Part of the confusion is that the disctincion between i and j is still comparably young. The manual for the Unifraktur Maguntia font states:
In some early fraktur texts, j was used at the beginning of a word
for both, i and j, while everywhere else i was used for both.
Early fraktur texts should be placed around the 16th century, so a clear distinction between i and j is only about 500 years old – with capital I and J remaining undistinguished in German blackletter type until well into the 19th century. Arguably, this is still older than the development of separate u and v letters but equally arguably the latter are more clearly distinguished in speech.
As for the names you have given, Jannine and Jessica with a /j/ sound at the beginning of the word are entirely possible. I somehow seem to recall having met a /j/-Jessica at one point in my life. Although it is probably true that the more prevalent forms are the ones with English pronunciation of the J.
It’s worth noting that the discussions above do not necessarily apply to loan words, especially if the language of origin uses the Latin alphabet. Thus, Jalousie and Journal are pronounced with French j while Junta is sometimes pronounced with an approximation of a Spanish j. And if one is talking about Poland’s Sejm, the j is never replaced by i even though pronunciation would suggest that.