Note that B31.3 also provides further thickness limits. Steve has given you the rational in his initial sentence.
In terms of the 5/8" minimum limit, it was instituted in the 1974 Edition. There are conflicting theories behind its imposition; one was that Section III (also fully revamped in 1974) only required impact testing above 5/8", so it was driven by the nuclear industry. The other is based on transference from 2 dimensional cooling to three dimensional cooling during normal welding conditions, so it was linked to heat input.