What is the difference between bath and bathe; born and borne?

It’s not just an additional ‘e’ but the difference in usage and meaning that makes these words different.

Bath is a noun which means washing of the body or the place for bathing in.

I’m going to take a bath.
I left my towel in the bath.

Bathe (pronounced as baid) is a verb which means to take a bath.

I was bathing when you called.

Born means given birth to.

He was born on 28th morning.

Borne means something carried or endured ( suffered bravely )

The child was borne on the shoulders of his father.
The hardships well borne fetch good results.