The TS1500 is a mavericky beast that was sold only in North America, as far as I know. In the scheme of transatlantic Sinclair models, it lies between the ZX81 (Timex 1000) and the original ZX Spectrum (Timex 2000 in the US). The case styling is almost identical to a 16K or 48K "rubbery" ZX Spectrum, except that it's made of a groovy silver plastic instead of the Spectrum's sedate black. The case even has speaker holes where the ZX Spectrum has them, although the TS1500 has no sound capabilities.
From a user's standpoint, the TS1500 is nothing more than a ZX81 (TS1000) with a slightly better keyboard (chiclet calculator keys instead of a dead membrane) and the 16K RAM pack included internally instead of being an add-on. Electrically, it is somewhat different from the ZX81 - the 40-pin SCL chip from the ZX81 has been replaced by a 68-pin PLCC custom chip from Timex. Oddly, the integration level seems to be lower in the TS1500; it has more parts than the TS1000/ZX81! The only functional difference I can find is that the RF modulator has software switching between channels 2 and 3 (it powers up on channel 2 by default; press 3 immediately after powerup and it switches to channel 3).
The machine was apparently a monumental flop, which isn't surprising. The unit I have was sold as part of an educational set, from some kind of educate-yourself-at-home university called "National Education Corporation". They sold the unit in a custom-fit plastic briefcase with the Timex/Sinclair 2022 tape recorder, Psion's Vu-Calc, and proprietary learning software, along with a bundle of thin silver booklets containing the course material.
This machine is fairly rare. I paid a little over US$100 for my mint-condition ex-teaching system (illustrated below) with its plastic briefcase, and I have reason to believe that was a rather good price. I only bought it out of a sense of collecting, however; if you want to immerse yourself in the early ZX experience, you would be better off buying a ZX81 or TS1000.
Please see the ZX81's emulator section for information on emulating the TS1500.