Usually Christians write He, His and Himself with a capital H when referring to God the Father. I'm not aware of this practice for either Jesus or The Holy Spirit.
Why the distinction?
Depends on the style guide, but it is a convention to capitalize names of God.
The Chicago Manual of Style does not require it, but does acknowledge it.
If a person doesn't capitalize He in regards to Jesus or the Holy Spirit, it is probably just an unintentional slip.
"Never to attribute to malice where simple incompetence will suffice."