So in the manga,
A yardrat is teaching Vegeta spirit control
How is this different from ki control? Supposedly ki is "the circulating life force", and spirit in many beliefs refers to our life force energy.
What's the difference between ki control and spirit control?
"Spirit Control" is an advanced form of ki manipulation.
At its most basic, we can perceive spirit control to be a sharper, more precise method to manipulate ki. In a way, spirit control is the Dodonpa to the ki control's Kamehameha : sharper, more refined, more efficient, but also more complicated to master.
The wiki states :
"Spirit Control allows its user to achieve a multitude of potent abilities through their ki, including Instant Transmission (a basic ability), Cloning, Gigantification and Healing (an advanced ability). These abilities achieved through Spirit Control are used by shifting, splitting and growing the user's spirit."
Some of these techniques are already Dragon Ball main-stays ! Healing has been regularly practiced by Dende and other Namekians Supreme Kais can also heal other individuals (Dragon Ball Super, chapter 20). Likewise, we have seen numerous other characters pratice Gigantification ; Piccolo became a giant during the final rounds of the 23rd World Martial Arts Tournament. Cloning was used by Tien Shinhan against Goku in the 23rd World Tournament (Dragon Ball, chapter 178) and against Hermila (Dragon Ball Super, episode 106).
Spirit Control appears to drastically improve ki flow for practitioners. Vegeta's ki output has been shown as massively stronger without actually tweaking the amount of ki released (Dragon Ball Super, chapter 55). The in-universe explanation is that the user's body is "synced up" with their mind, thus dramatically improving their ki manipulation efficiency.
You could consider Goku and Vegeta as extremely muscular men... who couldn't throw a good punch and efficiently use their muscle power before learning Spirit Control.
