Their previous lives are NOT wiped away.
Lord Commander Mormont is not saying you must give up your houses, as that is not the case for the Night's Watch. As TheLethalCoder said, he is merely saying that everyone comes as equal.
Some of you bear the names of proud Houses, others only bastard names or no names at all. It does not matter. All that is in the past. Here on the Wall we are all one House.
Outside the wall, those with the names of proud Houses would be considered superior to bastards which would be considered superior to those without any names. At the wall however, none of this matters, they are all equal regardless of previous rank.
As for why Jon was given a hard time for not giving Lord Slynt his proper title, those who have been knighted or given Lordship before swearing their vows at the Wall keep their titles. Lord Mormont was Lord Mormont at the wall before he became Lord Commander, similarly, Ser Alliser Thorne was knighted during the Targaryen rule and fought along side the Royalists during Robert's Rebellion
Alliser was a knight of House Thorne in the crownlands and fought on the side of House Targaryen during Robert's Rebellion.
A Wiki of Ice and Fire - Sir Alliser Thorne
Ser Alliser Thorne did not lose his title when he came to the wall, just like Lord Slynt didn't.
As for the words in the oath quoted by TheLethalCoder,
I shall wear no crowns and win no glory.
This simply means that they will not take any more crowns. They forsake their right to be the heir of their house or the Seven Kingdoms when they come to the wall. They can no longer return to their houses and take up the Lordship of their house. As for win no glory. They will not be raised for their glory in battle, they will gain no lands and no titles. They live to serve the Night's Watch and will die serving the Night's Watch.
The only orders where your House is removed his with the Maesters and the Faith, they only retain their first name, this led Maester Aemon to lead a forgotten life as the Last Targaryen living in Westeros. Otherwise, only the High Septon has to give up their name entirely. Although this seems to have been contradicted by the final episode of Season 7.