Consequences of BPP=BQP If BPP=BQP then there is a polynomial time randomized factoring algorithm.  A lot of other quantum algorithms that appeared to have an exponential speedup have recently been dequantized. For examples, see https://arxiv.org/abs/1910.06151 and references therein.

What other significant practical consequences would BPP = BQP be likely to lead to?
Assuming a practical algorithm is found, the applications will overwhelmingly be in quantum simulation. Basically any research in large quantum systems - chemistry, biochemistry, condensed matter, nuclear systems will be affected. We'd likely see major advances in basic biology, medicine and pharmaceuticals. Moore's law would be restored for a few years and we would finally have working fusion reactors. Physical simulation using the new algorithm might also lead to new physics.