Why did Don Draper steal the identity of his fellow soldier?

"What was the motive for switching identities?"

It was both to get a clean break from his past, and to escape the war.

Richard "Dick" Whitman took the identity of "Don Draper" (synonymous with "Wear Disguise") as soon as he realized the opportunity existed. By performing this transformation/exchange, Richard escaped his prior life with seemingly no chance of ever being discovered. This allowed Richard a free pass to follow a completely different life path, unburdened by his past. It has absolutely nothing to do with trying to assume the original Don Draper's actual past; Richard wanted nothing to do with Draper's prior life. He simply wanted a clean slate; a fresh start.

The death of Don Draper took place towards the end of his tour of duty, but near the start of Richard's own tour. By adopting Don's identity, Richard left Korea much earlier than he was supposed to, saving himself from the war.

By giving his family Don's body identified as his own, he provided them with a dead end, and Richard assumed that by disappearing, it would be easier to deal with Draper's family than it would be to deal with Whitman's family, if it ever came to it.

Don was never revealed to be Richard until (1) Anna Draper tracked him down privately, and (2) his younger half-brother found him during Season 1, having originally spotted Richard hiding when his parents received the casket containing the body of Don Draper, but being too young and powerless at the time to do anything about it.