Both “born” and “borne” are the past participle (third form) of the verb “bear”.

The verb “bear” basically means “carry”.

“Born” is used ONLY with reference to a birth of a baby or animal.

My son was born on Christmas Day.
He is a born leader. (This means he was born to be a leader.)
Many Cuban-born Americans live in Miami.

“Borne” is used in all other instances.

This issue must be borne in mind when considering future strategy.
He has borne the guilt of that financial loss for all these years.
Lyme Disease is a tick-borne bacterial infection.

Note that both “born” and “borne” can be used as adjectives as well as verbs.

In born leader, “born” is an adjective.
In Cuban-born Americans, “Cuban-born” is a compound adjective.
In tick-borne bacterial infection, “tick-borne” is a compound adjective.

Although I say that “born” is used only with reference to a birth, “borne” can also be used with reference to a birth if the reference is in the active voice.

In the sentence My son was born on Christmas Day, “was born” is the passive voice. This is the most common usage.

Other examples of this usage:

When were you born?
I was born in 1985.
My father was born before World War Two.
The twins were born three hours apart.

However, if we use the active voice we must use “borne”. Note that this usage is rather old fashioned.

Examples:

She has borne a son. (This means she has given birth to a son.)
Elizabeth had borne six children by the time she was 25.