In most contexts, start to V and start Ving have the same meaning, as in these two English translations of the same Russian short story:
And the custodian went up to the door and started knocking, but neither Trachtenberg nor Tinbergen opened to him. — Sasha Sokolov, Carl R. Proffer, trans., “Nymphaea,” A School for Fools, 1977.
And the maintenance man came to the door and started to knock, but neither Trachtenberg nor Tinbergen opened it. — Sasha Sokolov, Alexander Boguslawsky, trans., “Nymphaea,” A School for Fools, 2015.
While the choice may be purely stylistic, I sense a slight difference in topicality. When the custodian starts knocking on the door, he begins and most likely repeats the action; the sound of knocking reverberates in the readers’ ears. When the maintenace man starts to knock, I am more conscious of the time when he intiates the action and the expected outcome, which does not occur.
In her monograph The Semantics of English Aspectual Complementation (2012), Alice F. Freed points out that in contrast to begin, the verb start can also denote the onset of an action without its actual completion. To knock on a door, for instance, requires (1) approaching the door with the intention to knock, (2) raising the arm, (3) forming a fist, then (4) striking or tapping the surface of the door with the knuckles some number of times. If only the onset of the action occurs, i.e. anything or everything before (4), then the idiomatic choice can only be the marked infinitive, not the gerund-participle:
He walked up onto the big front porch and started to knock but noticed a button beside the door. He didn’t remember the house having a doorbell before he had gone into military service. — Charles H. Jackson, Death Hole Creek, 2012, 127.
In other words, the man at the door was about to knock on the door when he saw the doorbell button, so he used that instead.
In traditional terms to start knocking expresses the ingressive or inceptive aspect, while to start to knock without knuckles hitting the door is the prospective: he was going to/about to knock on the door, but then saw the doorbell button.