Subtle nuances, but:
- Dangling can be a transitive verb meaning "to make hang (from one's hand)" or intransitive, "to hang precariously".
- Hanging can also be transitive, but it would mean "to make hang (on something, e.g. a hook, rack, or rope)" usually "hanging up" in that case though, or intransitive "to hang (from something)".
- "Hanging on" implies someone is hanging from their own hands/arms and holding themselves up (not supported by say a hook or rack or something) or on to something.