Using the second benefit prevents any use of the feat on the same creature.
The description of the Healer feat's benefits is:
- When you use a healer's kit to stabilize a dying creature, that creature also regains 1 hit point.
- As an action, you can spend one use of a healer's kit to tend to a creature and restore 1d6 + 4 hit points to it, plus additional hit points equal to the creature's maximum number of Hit Dice. The creature can't regain hit points from this feat again until it finishes a short or long rest.
"This feat" includes both benefits, and both of them cause creatures to regain hit points. So the only reasonable interpretation of the last sentence of the last feat is that its use on a creature prevents either benefit (that is, "this feat") from causing that creature to regain hit points until they finish a rest.
In case it isn't clear, the resulting play pattern will be something like this, featuring the fighter Lucinda the Luckless and her pal, Harriet the Helpful.
Lucinda: Ack! That last hit dropped me, and then I took another arrow, so one more failed death save and I'm toast!
Harriet: No problem, I'll use a charge from my healer's kit to stabilize you, and also give you one hit point.
Lucinda: Cool! I'm back up! Take that, enemies! ... aaand I'm down again. Help pleeease!
Harriet: Again? Okay, one more charge, you are again stable and have one HP. Be careful out there!
Lucinda: Nice! Uh, I'm gonna disengage for a sec and hide behind that table. Wanna meet me there so you can give me more heals?
Harriet: Okay, sure, I'll move there, use the healing kit again ... have ten hit points.
Lucinda: Perfect! Okay, I'll jump over the table and wade into the fray again ... and I'm down. Haaaaariet, help please.
Harriet: (sigh) Alrighty, once more I crawl into danger and use a healing kit to charge to stabilize you.
Lucinda: You're the best! Back into the fight ...
DM: Nope, you're still at zero HP, prone, and unconscious.