The Sinclair ZX81 is based on the Sinclair ZX80 and the Timex Sinclair 1000 is the version of the ZX81 as sold by Timex, USA. They are quite similar.
The ZX81 is based on the ZX80. All standard TTL ICs of the ZX80 were combined into one ULA (user-designed logic array).
The Basic ROM was extended from 4 kB to 8 kB. While the ZX80 Basic could only handle integer numbers, the ZX81 Basic now could handle floating point.
The video display procedure was improved. The ZX80 could only either generate a TV signal or execute a useful program, while the TV was left black. This resulted in the TV flickering every time a key was pressed and needed to be handled and the screen blanking while a program was processed.
The ZX81 also was capable of this mode, now called the FAST mode, but also had an additional method to display TV and execute a program virtually at the same time. In this mode the TV no longer flickered when the user pressed a key. It was called SLOW mode, because the CPU still needed to execute special code in order to display the TV image. Only during the vertical flyback of the electron beam the CPU executed the user program. So in this mode the ZX81 was effectively as slow as a 900 kHz throttled Z80 computer without this burden upon the CPU.
For the SLOW mode an NMI (non-maskable interupt) was used to count TV lines. Though the NMI could have been used for this purpose during PAUSE and INPUT in FAST mode too, Sinclair kept the code from the ZX80, which did not yet use the NMI. Due to this nice move, the ZX81 ROM could be used in the ZX80. You only couldn't switch to SLOW mode on the ZX80, because of the missing NMI generation by the ULA. There were simple circuitries published which added the NMI generation to the ZX80, making it fully compatible with the ZX81. On the other hand you could insert a ZX80 ROM into the ZX81 and disconnect the NMI line, thus downgrading it to a ZX80. :-)
The ZX81 and the Timex TS1000 are nearly identical. Besides a different logo they differ slightly in the TV signal generation logic.
The ZX81 generates the TV signal at UHF channel 36 with 50 frames per second.
The Timex 1000 uses a different modulator and emits the TV signal with 60 Hz on VHF channel 2 or 3 – selectable with a switch in the bottom of the case. The different channel is accomodated with a different modulator, which – what a nuisance! – has it's TV socket on the other side, making the top case of the ZX81 and Timex 1000 incompatible. The higher refresh rate of the Timex is produced by the display program executed by the Z80 CPU. For 60 Hz it generates less lines above and below the usable screen. Whether to use 50 or 60 Hz is read each frame from an I/O port which is connected to a wire strap on the PCB – the ROMs are identical. The PCBs are identical too, only some optional components are added or replaced for the TS1000.