1 He lives off/sponges off his parents because he doesn't have a job.
2 Teenagers live off their parents till they are 18.
3 He is a lazy bones. He doesn't want a job because he sponges off his parents.
In this context, the two have basically the same meaning. The latter is more negative than the first, but the first is not positive.
Both phrases can have different meanings in different contexts.
John is an organic farmer and he lives off the land.
Mary lives off campus.
The Wilsons live off Highway 101.
The pediactrics nurse carefully sponged off the baby.
A chamois was used to sponge off the water.
MarK went fishing for sponges off the Florida coast.
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