On paper, they’re both marketed as “Analogue Drum Machines” and they are both similarly priced but in reality they are two entirely different units with neither being entirely analogue. Without going into specific features of each (which you can find on the manufacturers websites and countless other sources), the choice simply comes down to whether you want instant gratification or if you want to get into deep synth style programming.
Both units are 100% capable of pumping out solid beats, there is no question about it but the approach, sonic variety and time investment differs due to the architecture of each machine’s sound generation design.
The Tempest approach is geared around using classic subtractive synthesis layered with ROM (read only) samples. The architecture for each drum sound/voice is identical so it’s really up to you to program it to your liking in order to shape the drum sound that you’re after.
The Rytm on the other hand, is set up with fixed architecture for each type of drum (kicks, snares, hats etc) so that you already have the basic architecture of that drum programmed to emulate the sonic character of it. This is a huge time saver at the expense of flexibility. However, the Rytm also allows you to load samples which greatly expands the tonal palette.
Both units offer excellent performance and sequencing features so the main difference between them comes down to whether you want instant drum machine gratification, in which case you should grab the Elektron Analog Rytm, or if you want to get right into sound design from scratch, which is Tempest territory!