Over evolutionary time, the ability of individual genes to perform their function climbed through genomic space to local fitness optima. For basic biochemistry, most of this was accomplished in deep time during the evolution of archaea, the bacterial symbiotes that became our mitochondria, and early eukaryotes. We share a huge amount with yeast (last common ancestor, perhaps 1200 mya). Of 450 genes critical for yeast survival in laboratory conditions, about half can be swapped out for the human version and the yeast can survive and reproduce.
I think with model organisms, its useful to consider the time of our lineages' divergence:
Escherichia coli: 3-4 billion years ago
Saccharomyces cerevisiae (Baker's yeast): 1200 million years ago
Caenorhabditis elegans (nematode worm): 560 mya
Drosophila melanogaster (fruit fly): 560 mya
Danio rerio (zebrafish): 430 mya
Xenopus tropicalis (Western clawed frog): 340 mya
Canis lupus (wolf, domestic dog): 85 mya
Sus scrofa (domestic pig): 85 mya
Rattus norvegicus (brown Norway rat): 65-75 mya
Mus musculus (house mouse): 65-75 mya
Macaca mulatta (rhesus monkey): 25 mya
Pan troglodytes (chimpanzee): 7 mya
In the grand scheme, we and mice are fairly near cousins. With mice, at the genetic level we share pretty much everything but developmental tweaks and hundreds of olfactory receptors that our primate lineage lost. Our genes can generally be freely swapped with mice versions, and the mice are still viable. This is how hundreds of mice models of human diseases have been produced, everything from ApoE-/- to Huntingtons.
IMO, compared to mice we certainly differ in the extent to which we're effected by environmental/diet/exercise interventions, but usually not in the direction. If something benefits mice its generally neutral to beneficial in humans, if something harms mice its neutral to harmful in humans.