Since their initial inventions (Rife - Rife Machine, and Clark - Zapper), there have been numerous enhancements and or knock-off/spin-offs from these original designs, so it's a somewhat merky topic to clear up, but I hope my comments help.
The quick and simple answer is - Rife Machine and Clark Zapper have the same basic goal that is achieved through different means.
As far as timeline goes, Rife came before Clark; in fact, a lot of Hulda Clark 's R&D effort for her first zapper design was based upon the microbe/frequency chart that Rife had discovered in his R&D many years prior. Rife developed a chart for the various frequencies required to rid the many individual pathogens that exist in the human condition.
The Rife - Rife Machine, and the Clark - Zapper (first generation) had at least one principle in common, and one principle that was significantly different. As for the former, both designs were made to take advantage of the MOR - mortal oscilitory rate, another Rife discovery of the frequency of signal required to disable a particular pathogen. As for the latter, the Rife Machine achieved it's beneficial effect through EM (electro-magnetic) radiation, whereas the Clark Zapper achieved it's effect by direct contact with a much weaker signal. In other words, with a rife machine generating upwards to hundreds of watts of RF output, a patient only need to be near, or "in the neighborhood" of the machine to recieve it's beneficial effects, in this case I believe one needed to be within several yards of the device, but not touching; kinda like being near a radio or TV or Cellphone or Microwave tower, except in the case of the Rife Machine, one is willingly being bombarded by a signal with desired and beneficial effects, but in the case of Radio/TV/Cell/Microwave, one is often unknowingly and unwillingly being bombarded by signals with malevolent effects ;) The zapper, on the other hand, has a much lower output (very few watts, low voltage, micro-amp current), therefore the need to be in direct contact with it (IE> the "hand-holds"). The Rife Machine (I believe) can be dialed into any frequency, like a frequency generator, to target specific pathogens. The first-gen zapper only outputed 30,000 Hertz, but with the resultant "harmonics" is able to zap pathogens across a somewhat broad spectrum.
Keep in mind that frequency generally implies "AC" but in the case of the zapper, the frequency is actually derivied as a result of positive DC beign turned off and on 30,000 times a second, whereas the Rife machine truly uses an AC-based output signal.
Here is just one of many links out there for Rife info.