Why do LEDs have/need positive and negative nodes to function whereas say resistors don't?
Different devices are different. It doesn't matter which end you hold to swing a stick, but it does matter if it is an axe or baseball bat. Whats the difference? Symmetry.

For a diode, one end of it is made of P doped silicon and the other end is N doped silicon. A resistor is just the same material throughout. If the component is not symmetrical then it stands to reason it won't behave symmetrically either.