Would rather vs. would prefer - to express a specific preference

When we speak about a specific preference,would rather and would prefer have the same meaning and are interchangeable.

We went to the theatre yesterday. Today Iwould rather go to the cinema.
We went to the theatre yesterday. Today Iwould prefer to go to the cinema.

Would rather can be abbreviated to 'd rather.
Would prefer can be abbreviated to 'd prefer.

I'd rather go to the cinema today.
I'd prefer to go to the cinema today.

Would rather is followed by the bare infinitive. 
Would prefer is followed by to + infinitive.

I'd rather have water.
I'd prefer to have fruit juice.

We use a past tense after would rather when we speak about the actions of other people, even though that action may be in the present or future.

I'd rather you took a taxi than walk - it's not safe on the streets at night. 
The film is quite violent. I'd rather our children didn't watch it.

When making a comparison we say:
would rather ... than

It's such nice weather - I'd rather sit in the garden than watch TV.

After would prefer we say:
would prefer ... rather than 

It's such nice weather - I'd prefer to sit in the garden rather than watch TV.

Prefer vs. would rather - to express a general preference

When we talk about general preferences we can use prefer or would rather. The meaning is the same.

I prefer walking to cycling.
I'd rather walk than cycle.

After prefer we use the verb in its ing form.
After would rather we use than.

I prefer using a keyboard to writing with a pen.I'd rather use a keyboard than write with a pen.