The difference between Hold on and Remain

When used as verbs, hold on means to grasp or grip firmly, whereas remain means to stay behind while others withdraw.

Remain is also noun with the meaning: that which is left.

check bellow for the other definitions of Hold on and Remain

Hold on as a verb:

To grasp or grip firmly.

Examples:

"'Hold on tightly to the railing."
Hold on as a verb (idiomatic):

To keep; to store something for someone.

Examples:

"'Hold on to my umbrella while I ride the roller coaster."
Hold on as a verb (idiomatic):

Wait a short while.

Examples:

"'Hold on while I get my coat."
Hold on as a verb (idiomatic):

To remain loyal.

Examples:

"He didn't give up his fandom when others did; he held on."
Hold on as a verb (idiomatic):

To persist.

Remain as a noun (mostly, in the plural):

That which is left; relic; remainder.
Remain as a noun (in the plural):

That which is left of a human being after the life is gone; relics; a dead body.
Remain as a noun:

Posthumous works or productions, especially literary works.
Remain as a noun (obsolete):

State of remaining; stay.

Remain as a verb:

To stay behind while others withdraw; to be left after others have been removed or destroyed; to be left after a number or quantity has been subtracted or cut off; to be left as not included or comprised.
Remain as a verb:

To continue unchanged in place, form, or condition, or undiminished in quantity; to abide; to stay; to endure; to last.

Examples:

"There was no food in the house, so I had to remain hungry."
Remain as a verb:

To await; to be left to.
Remain as a verb (copulative):

To continue in a state of being.

Examples:

"The light remained red for two full minutes."