Half time = half life
Both terms refer to how much time will pass until 1/2 of the substance has been cleared from the site of action (by metabolism, excretion, or redistribution). "Elimination" usually implies that the drug is gone (metabolized or excreted, never to return) but there's some ambiguity to the term.
What's more relevant to anesthesia is context-sensitive half-time, which is generally unrelated to elimination half-times and increases as the duration of an infusion increases. Ie, a 50 mcg dose of fentanyl will last a lot longer if the patient has been receiving an infusion for a while.
The reason for this is that the effect of a drug is a consequence of its concentration at its site of action (some receptor), which is closely correlated to the value we can measure, plasma concentration. Drugs "wear off" because they are metabolized, excreted, or redistributed to tissues where they are not active. It's the redistribution and sequestration of a drug, with eventual re-entry into circulation, that complicates the duration of an observed effect.
The classic example for this is comparing narcotic infusions:
The differences are mainly explained by differing volumes of distribution.
Fentanyl is very fat soluble, and initial doses "wear off" relatively quickly partly because the drug redistributes to adipose. If an infusion is ongoing, it doesn't take long before peripheral compartments have a large store of fentanyl, which rapidly replenishes any drug that is metabolized from the plasma. Consequently, fentanyl's curve gets very steep, very quickly, and later doses last much longer than the initial doses.
Remi is flat because its efficient metabolism by nonspecific plasma esterases renders its half-time more or less constant regardless of dosing or infusion length (at least, for clinically relevant numbers).
This graph isn't totally accurate; Sufentanil doesn't really flatten out like that after just a few hours. It continues to increase, but nowhere near as steeply as fentanyl. Its volume of distribution is gigantic so even after a long infusion is stopped, redistribution still accounts for part of the decrease in plasma concentration.
"Depleted" doesn't mean "totally 100% gone" ... tachyphylaxis refers to a
reduced effect from a subsequent dose of a medication, not
no effect. To an extent larger doses can be given to achieve the same effect as earlier doses. The utility of endlessly increasing subsequent doses has a couple of limitations though -
1) eventually there's just no more effect to be had regardless of dose
2) toxicity / side effects (nitroprusside, for example)