Signal processing in FMCW and SFCW
You can build receivers and transmitters for both digitally covering the whole band, or just the beat frequency.

So, there's no "general" difference; you'll need to look at actual implementations.

There's different motivations that lead to people adopting either: in analog systems with expensive VCOs, it's easy to do a linear chirp to modulate and demodulate with simultaneously; in digital frequency synthesizing, that might not be the case.

But: there's plenty of reasons you might want to e.g. do the analog sweep on transmit side, but don't mix down with the same (for example, bistatic radars, or radar variants where you want to observe very high doppler); or use a stepped frequency on transmit side from a discrete-frequency synthesizer, and still do the mixing in analog.

In the end, it's all a complications + costs vs benefit tradeoff: for some systems, digitizing with a high bandwidth is easier than mixing down with the actual transmit LO, for others it's not.

However, it's extremely rare that you directly sample the RF signals (maybe barring a few late cold war over-the-horizon radars): mixing down to baseband or at least an IF, or to beat frequency is cheap, and radars typically work at relatively high center frequencies.