For a body in orbit in 2-body scenario, for Keplerian elements only a single variable (true anomaly) changes over time. For Carthesian the whole state vector is in constant flux.
When working with orbital mechanics in Keplerian coordinates, e.g preparing maneuvers, you don't need to work on a body in motion - 2-degree differential equations of motion all the time, as typical to most of Newtonian mechanics and similar. You just stash time (and true anomaly) temporarily forgetting about them, and work out the completely time-independent, static geometry, finding the needed maneuvers and trajectories. When you have the entire plan ready, just retrieve your time/true anomaly data, substitute values and get the moments of maneuvers.
It also means in Keplerian mechanics, the vector state holds quite explicit history and future of the position, besides position at the moment. Things like eccentricity or inclination tell a lot about the orbit at a glance. Meanwhile, given a velocity and position vector of a satellite, and mass of the central body, telling whether it's nearly on escape trajectory, or in equatorial circular orbit takes a good bit of calculations.