leaving something in an improper state
Attaches to the -masu stem of a verb. Indicates that something is left undone, open, on, or in an undesirable state, often with a nuance of neglect or carelessness.
Please don't leave the door open.
continuously; keep doing
Attaches to the -masu stem of a verb. Describes an action continuing without pause, often with a sense of relentlessness or excess.
It has been raining nonstop since yesterday.
Please don't leave the water running.
He left work unfinished and went home.
He kept talking for a whole hour.
My legs started hurting from standing all day without a break.
The TV had been left on.
まま means 'leaving something as it is' and attaches to nouns or past-tense verbs with a neutral tone, while っぱなし attaches to a verb's -masu stem and usually implies careless neglect.
続ける is a full verb meaning 'to continue,' whereas っぱなし is a suffix on a verb stem that adds a nuance of relentless or excessive continuation.
From 放し, the continuative form of 放す, used after a verb stem for leaving an action or state unchecked. A preceding small っ commonly changes はなし to っぱなし.