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.