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不 and 無 are negation particles. So why don't 不由 and 無由 mean the same thing? 不由 appears to be the snag, because its meaning isn't the negation of 由. If you 不由 Φ, you still do Φ,!
ABC Chinese–English Comprehensive Dictionary (2003), p 74.
Op. cit. p 1013.
(iii) privative polarity: one term is associated with the presence of something salient, and the other with its absence.
Alan Cruse (died 2020), Meaning in Language (2011 3e), p 163.
不由 has two main ways of being used, both implying you cannot avoid doing X.
the first implies that you cannot avoid something as a result of a cause already mentioned. example sentence : 你劳得这么艰苦了，不由得我不表扬。 You worked so hard, I couldn't not praise you. (cannot but praise you)
The second use of 不由 is that you do something involuntarily or against your will. Example sentence: 我太累了，不由打呵欠。I was so tired, I couldn't avoid yawning. (can't help yawning)
Meanwhile 无由 implies that there is no other option, it isn't a matter of being forced to choose but that you never had a chance. example sentence: 我需照护我妈妈了，无由上学。 I need to take care of my mother, it is impossible for me to go to school. (have no way to go to school)
As for why they don't have the same meaning as each other 不由 implies other options are available and 无由 is used (uncommonly) when there was no chance of something else happening.
Unfortunately languages are not designed like software code with a rigorous and consistent logical structure, but are evolved through how people always speak.
Perhaps not the answer you are seeking, but they mean different things because they just do, with no extrapolatable, extensible, universal logic to be applied in other Chinese language scenarios.
However, if you are to find a reason, 无 generally would imply 'absence' of something, or 'nothing', whereas 不 is more negation. So in this sense 无由 would be absence of reason (无理由）, and 不由 would be can't help.
