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Fourier-based techniques (FBT) and sample slicing (SC) may be similar if doing "raw" transformations, but FBT can potentially be cleaner, or at least easier to clean up. If you use raw "bit-maps" for FBT, yes it will be choppy like SC, but one can use regression or regression-like curve-fitting to give FBT smooth time/frequency curves to synthesize against, sounding more natural. There are down-sides to using regression, but for typical voice and music, those won't matter much.

One rough area for curve-fitting is white-noise-esque sounds (WNES) like the letter "s" or "h" and tambourines. The processor can perhaps detect if WNES exceed a threshold, and use other techniques such as SC instead.

It's roughly comparable to JPEG versus GIF images. JPEG is better (more faithful) at gradual shades while GIF is better at edges. A better compression algorithm perhaps would use each where it does best per given image. However, at the cost of algorithm complexity and compression/decompression processing time.