Depending on the nature of the post-processing, 24-bit could sound significantly better, even with 16-bit source material. Digital audio has limited precision, depending on the number of bits. The greater the bits, the greater the precision.
When you sum multiple sources, you effectively lose precision from each input unless you increase the number of bits used to represent the output. So by mixing to 24-bit precision, you aren't losing as much quality at that phase.
When you then do the final normalize and process it down to 16-bit 44.1 kHz or whatever your final output format is, you'rel basically taking a 16-bit-tall slice out of the... maybe 22.5 bits of dynamic range that you actually used, and then representing that slice as 16 bits in the output. The result is a full 16 bits of precision in the output stage. That's a lot better than you would if you mixed down to 16 bits of precision initially, then stretched the 14.5 bits of actively-used precision into 16.
Of course, if your output is perfectly normalized at 16-bits to begin with and you aren't doing any post-processing, then the issue is moot, as the results should be exactly the same, assuming that whatever you use for post-processing handles the down-conversion in the same way. In that case, it's just a question of which tool does the rounding/truncating/dithering.