Rather Comparative preference

Prefer X to Y

WOULD PREFER TO			WOULD PREFER…RATHER THAN
We express preference with would prefer X to Y. Parallel (syntactically alike) wording is used.			Speakers tend to switch from prefer X to Y to prefer X rather than Y when the options consist of infinitives or that-clauses. (MWDEU 760)
SUBJ + VERB I would prefer	OPTION 1 walking	OPTION 2 to  driving  (gerunds)	SUBJ + VERB I would prefer	OPTION 1 to walk	OPTION 2 to  to drive    (Reword it with rather than.)
I would prefer	tea	to coffee  (nouns)	I would prefer	to walk	rather than  to drive.
I would prefer	red	to blue. (nouns)	I would prefer	that we walk	to  that we drive.
I would prefer	going in the morning	to (going) in the evening (gerund clause)	I would prefer	that we walk	rather than  (that we) drive.
I would prefer	outside the restaurant	to inside the restaurant (prep. phrase)	I would prefer	to be done	rather than to be working.

Would rather X than Y

WOULD you RATHER X OR Y			WOULD RATHER X THAN Y
We use would you rather X or Y? to ask preference between two items.  Parallel (syntactically alike) wording is used before and after to.  (Or functions as a conjunction.)			Would rather than also coordinates two options phrased with parallel wording.  (When parallel phrasing is used than is more conjunction. When non parallel phrasing is used than is more preposition.)
AUX + SUBJ Would you	OPTION 1 rather drive	OPTION 2 or walk?  (parallel adjectives)	SUBJ + AUX I would	OPTION 1 rather drive	OPTION 2 than walk.
Would you	rather leave at 8:00	or wait until 9:00?	I would	rather leave at 9:00	than 8:00.
Would you	rather that I call you	or that you call me?	I would	rather (that) I call you	than (that) you call me..

Also see would

Comparatives Than vs. Rather than

THAN — AUXILIARY COMPLEMENT	RATHER THAN — NONFINITE COMPLEMENT
After the comparative use of than we use the auxiliary verb.	After the comparative use of rather than we use a nonfinite verb form (a "secondary" verb form that is not marked with tense.)
SIMPLE COMPARATIVE  – keeps auxiliary after than (parr all el reduced clause) He dislikes traffic more than you do.	MODAL TENSE – keeps nonfinite verb after than He will "thumb" a ride rather than [he will] pay the bus fare.  (thumb – hitchhike) [X and not Y]
He feels pain more intensely than everyone else does.   (verb remains from a reduced clause)	He'd rather leave at 4 a.m. than  [he would ] get stuck in traffic. (stuck – not moving) [X to avoid Y] (Getting stuck in traffic is something that drives him crazy.)