Hyperthreading is Intel's marketing name for SMT aka Simultaneous Multithreading - one physical core presenting itself as multiple logical cores, to help keep hungry execution units fed with work, especially for code that tends to stall or have other throughput bottlenecks (like dependency chains). See http://www.lighterra.com/papers/modernmicroprocessors/ for background info about modern CPU pipelines.
(In some CPUs, especially i5 series, Intel disables this feature for market-segmentation reasons. And the efficiency cores in Alder Lake don't support SMT at all, only the P-cores, so some chips have a logical core count that's not 1x or 2x the physical core count.)
AMD Zen also uses SMT in essentially the same way as current Intel CPUs, unlike Bulldozer-family. Wikichip's Zen article has an SMT section which describes some of the CPU architecture implementation details, like which structures are partitioned (split in half when both logical cores are active, not sleeping) vs. replicated vs. competitively shared.
See David Kanter's deep-dive into Bulldozer for some details about its CMT - "Clustered Multi-Threading", which is what AMD called their design of having two weak integer back-ends sharing a SIMD/FP unit and some resources. It had the major downside of not being able to run a single thread fast, only two threads with somewhat good total throughput. (Compared to Bulldozer's generally lowish performance, thanks to that and some other design decisions which AMD wisely scrapped for Zen, after having the chance to see how they panned out over the years.)
Intel's current incarnation of SMT is pretty similar to when they implemented it in Nehalem; see David Kanter's Nehalem deep-dive for more on that.
https://www.realworldtech.com/alpha-ev8-smt/ is a good intro to the SMT concept; Alpha EV8 was planned to be the first commercial CPU with SMT working the say it does in current out-of-order exec CPUs with it. DEC cancelled it late in the design process. Intel hired much of the design team who went on to re-implement SMT in Pentium 4, the first time the Hyperthreading marketing name popped up.
"Multithreading" in a CPU-architecture context includes SMT, but also coarse-grained things like "switch on stall" or in-order barrel processors. So it's a much broader term; there's a Wikipedia article that covers many of the strategies.
"Multithreading" in computer science in general is more about operating-system and software-level stuff.