Keep doing = to continue or cause something to continue in a certain state, condition or position; do repeatedly.
Keep on doing = to keep continuing/trying to do or persist in doing something without any intention of stopping.
The meanings of ‘keep’ and ‘keep on’ are almost the same and both can be used in most of the situations interchangeably. However, there is a difference that governs whether or not to use ‘keep’ or ‘keep on.’ Keep on is a phrasal verb which is an intensive version of plain keep in this sense:
He keeps on asking for more gravy.
She kept on speaking at the concert despite many disruptions from the audience.
Freddie kept on walking, though his feet got really tired out and rough.
The poor beggar paid no attention to her shouting and kept on rummaging through the garbage for his daily survival.
When we use keep on in the above sense, it becomes an intensive version of ‘keep’ that means to continue in a state, condition or position, and the ‘on’ is meant for emphasis.
Children kept on interrupting the grandma, so she couldn’t finish the story. (Children continued to interrupt the grandma)
When he got home he spoke to his wife and she kept on talking about the girl next door.
The ice cream man kept on playing the music until the kids came from neighbourhood.
She was sitting by his side on the couch while he kept on watching reruns of an old show.
In another sense, if you keep on doing something, you are usually a bit more stubborn than someone who just keeps doing something:
The teacher kept advising him to work hard.
Mother kept on advising him not to mix with the wrong people in the vicinity.
‘Keep on’ is also used in idioms:
‘Keep on keeping on’ means keep trying; keep doing what you are doing.
Do your best. Just keep on keeping on.
Keep on keeping on, Tom, don’t give up. Just one bad grade in school isn’t the end of the world.