Thrawn saw that Vanto had a natural aptitude towards tactics and strategies, and helped him develop those abilities over the years. However, Vanto was not a "peer" to Thrawn anymore than Nightswan was. The novel made it abundantly clear that Vanto was still far inferior to Thrawn as a tactician.
As for why Thrawn sent Vanto to aid the Chiss Ascendancy, I can only assume it was for the same reasons he mentioned to Nightswan, when he offered him the same opportunity:
IOW, there are FAR more dangerous threats in the galaxy than even the Empire, and the Chiss actively oppose them. Thrawn felt that Vanto(who Thrawn considered a 'master' tactician among humans) could be best utilized by the Ascendancy to help them fight these threats.