The point of the switch is twofold, I think.
The switch is only known to a very small number of people, while everyone else thinks that
Rand is dead.
Additionally, almost no one left alive
knows what Moridin actually looks like,
so no one will recognize the new form, and will just consider him to be a random person. Since
Rand no longer "channels" to channel, he should never be "detected" doing anything unusual, and
he can now go about complete unrecognized by the rest of the world.
One of the key aspects of the Prophecies of the Dragon that's been quoted at us for a long time was that
Rand had to die.
It was pretty explicit in other forms of prophecy as well, such as Min's visions and Egwene's dreaming. By the end of the series,
Rand did die, or his body did, it was just Moridin in the body at the time.
That allowed prophecies and visions to be fulfilled, just not in the way that everyone expected.