First, the concepts these two have of the 'functions' (socionics calls them information elements) are different, but related. Think of them as different takes on roughly the same ideas.
But on another note, I'd distinguish MBTI's test from the functions theories surrounding -- I'd say socionists generally would call INTJ on the TEST closer to INTj in socionics, but INTJ in the functions sense (NiTe) closer to INTp.
For what it's worth, I think the INTJ test being ~ INTj is closer to Jung's interpretation, in the sense that I'd say P on the test is more associated to ideas Jung used to characterize irrational types, and J more associated to ideas he used to characterize rational types. However, P~irrational and J~rational is also far from what I'd call a very neat identity or anything.
It's just I think the J ~ Je and P ~ Ji is not a very good idea. Where it roughly starts off, as far as I can tell, is that P does, all things considered, tend to do their own thing at exclusion of some standardized rule-system, so it's not like it was totally not understandable an association, but still, ultimately where I think this goes wrong is that there's this idea that the IPs (in the dichotomies sense) pursue some internally determined highly structured approach due to Ji....just because they don't pursue an externally determined structure but that's just a fallacy: it's more accurate to say they are overall less structured and more spontaneous on every count.
It's not that the idea of internally vs externally determined structure is necessarily not an interesting idea (that's separate a debate) -- rather, there's just no reason to think the MBTI test's measured dimensions really track that divide all that closely.