Other boards leave out input signal controls that real oscilloscopes have. The PC is used as an interface to display the data that was captured and stored on the high speed circuitry of our instrument. A faster PC will draw the screen faster, but it won't affect sampling rate. Our instruments have high speed sampling and data buffer circuitry that allow them to sample and record the data at high speed and then to transfer it to the PC after the data buffers are full. With longer memory, you can both maintain high sampling rates, giving you better waveforms, and see longer duration events in their entirety.
Memory is used to store the data that represents the input waveform.
Consequently, the longer the memory, the longer the period of time you can acquire. Or, more samples can be taken within the same period of time at a faster rate. Often, the known error occurs some time after the actual cause. Longer memory allows you to trigger on the known error, but also capture the sequence of events back to the root of the problem. A long buffer allows you to see both the problem and its cause.