The cult successfully manufactured complex chemical weapons, and had plans in place to manufacture nuclear and biological weapons. These are all weapons that can be difficult for state actors to produce, much less a fringe cult. How were they able to reach this level of sophistication? How did this amount of advanced technology in unqualified hands not set off alarm bells? Why have other large terrorist organizations not been able to reach similar levels of sophistication?
Currently writing a paper on Aum, so I have some of these answers, but the TL;DR is mass negligence on behalf of the Japanese authorities. This was compounded by the Japanese government promoting Aum to run a Russian university, where they acquired a bunch of weapons.
Basically, Aum managed to connect with highly educated people, who formed the core of the New Religious Movement. Endo Seiichi and Tsuchiya Masami had both trained as chemists, and were the ones primarily responsable for the manufacture of both chemical and biological weapons. They didn't do a fantastic job, as the sarin was only 30% pure - still enough to do the job though. This was because they weren't ready to go on the sarin attack - they were still in the development and testing phase.
They weren't caught in their other chemical and biological attacks despite Asahara signalling the groups intent in a speech, mostly because the guy putting together a TV expose of Aum was murdered by the group, and, as he was a Communist, the Japanese police didn't bother to investigate. Aum's status as an NRM was also a factor - the authorities didn't want to come down too hard on the group and be seen as repressing people's religious rights.
Sources
Ian Reader, Religious Violence in Contemporary Japan
Robert J. Lifton, Destroying the World to Save It
Why have other large terrorist organizations not been able to reach similar levels of sophistication?
It is worth noting that many large terrorist organizations do have significant numbers of well-trained people working in them. This is especially the case in the Middle East, where there is (because of Cold War policies) an overabundance of engineers relative to the number of jobs for them. Many of the high-ranking figures in al-Qaeda, for example, have extensive technical training, and engineers are over-represented among terrorists versus the general population in the Middle East.
See Gambetta and Hertog, Engineers of Jihad: The Curious Connection between Violent Extremism and Education (Princeton University Press, 2016). There was a nice write-up in Foreign Affairs a few years ago by the same authors.
What Aum Shinrikyo had was a potent mixture of access to specialists, access to capital, and access to the infrastructure available in a highly-technological nation. Groups like al-Qaeda and ISIL have (so far) had access to a few of those components, but not all of them. It is fortunately harder to do this kind of native development of WMDs (e.g., as opposed to stealing it) than most people realize.