One is a term; the other is a concept with a literature behind it. If you want to read further, search object constancy — it's what emotional permanence is describing, and it has clinical writing going back to Hartmann in 1952 attached to it rather than none.
Side by side
Aspect
Emotional permanence
Object constancy
Origin
Internet, unverified
Hartmann, 1952
Literature
None under that name
Substantial
In manuals
No
No — but it's a real construct
Useful?
As a description
As a framework
What each one means
Emotional Permanence
Emotional permanence is an informal, internet-coined term for trusting that someone's care still exists when they're absent, silent, or you're in the middle of a fight — and it is not a research term.
In psychoanalysis, the capacity to keep a stable, whole picture of someone — and stay connected to your good feelings about them — when they're absent, disappointing, or you're angry with them. In perceptual psychology the same phrase means something else entirely: seeing an object's size, shape and colour as constant even as distance, angle and light change.