"It is permitted to know" and "it is permitted to see," while literally correct translations of the two words in question, aren't actually what they mean.
"Latin commitment-markers: scilicet and videlicet," by Josine Schrickx, has a very detailed account of the similarities and differences between the two words, the gist of which is that scilicet means something like "of course" or "naturally" or "obviously," indicating that the speaker takes it as a given that the listener will agree with what s/he is saying, while videlicet means something like "evidently" or "apparently" or "it seems," indicating that, while what the speaker is saying is supported by the evidence, s/he is unwilling to commit to it as true.
Scilicet and videlicet are both evidential commitment-markers – the commitment is based on evidence – and have a comparable origin (from a form of scire / vidēre and licet). As we have seen, however, there are important differences:
scilicet indicates that the evidence is based on expectation (‘as is to be expected’, ‘of course’) and is strongly directed towards the addressee, videlicet indicates that the evidence is inferable from the context or reasoning
(‘clearly’) and is not directed towards the addressee. The differences are shown by the different contexts in which they occur: scilicet often appears with a verb in the first person, videlicet rarely so and if so then only when used ironically; scilicet can be used in ‘bases’ and contrasts, videlicet cannot; scilicet can be combined with a future form of the verb in the second person, to mitigate an appeal to the addressee, videlicet cannot; videlicet is rarely used with a future verb form of any kind; scilicet can be used independently as a reaction, videlicet cannot (or at any rate is not so used in our material). Both adverbials can be used ironically, in which case they are not strictly speaking commitment-markers. The speaker uses scilicet to appeal to the addressee and to show solidarity with him; it can be used for ‘positive politeness’.
Videlicet cannot be used for these purposes; with this marker speakers show that they have good evidence for what they are saying and do not need to appeal to the addressee for support or confirmation of the common ground.
Note that this article (and my answer) deal with how the words are used in Classical Latin. It's possible that their use shifted in later Latin and/or in English, but if that's so, I suspect it's a matter for a different question.