A word I haven't seen for a long time. Seems hardly to be distinguished from "exist".
But checking the dictionary shows there are some valid differences in other contexts.

subsist
v : support oneself; "he could barely exist on such a low wage";
"Can you live on $2000 a month in New York City?"; "Many
people in the world have to subsist on $1 a day" [syn: exist,
survive, live]


Source: Webster's Revised Unabridged Dictionary (1913)

Subsist \Sub*sist"\, v. i. [imp. & p. p. Subsisted; p. pr. &
vb. n. Subsisting.] [L. subsistere to stand still, stay,
remain alive; sub under + sistere to stand, to cause to
stand, from stare to stand: cf. F. subsister. See Stand.]
1. To be; to have existence; to inhere.

And makes what happiness we justly call, Subsist not
in the good of one, but all. --Pope.

2. To continue; to retain a certain state.

Firm we subsist, yet possible to swerve. --Milton.

3. To be maintained with food and clothing; to be supported;
to live. --Milton.

To subsist on other men's charity. --Atterbury.
Subsist \Sub*sist"\, v. t.
To support with provisions; to feed; to maintain; as, to
subsist one's family.

He laid waste the adjacent country in order to render
it more difficult for the enemy to subsist their army.
--Robertson.