If you want your new and old Facebook accounts to remain separate, you have to take extraordinary measures. Facebook's whole thing is about making connections between people, and its software is extremely good at it. It can tell from your Internet address that both accounts come from the same computer, so it can easily guess that both your selves know each other. Therefore all your friends on both accounts will be invited to 'friend' the other account.
You can keep two accounts separate if: you only access each account from a different computer; and you ensure that the two accounts have no friends in common, don't belong to or 'like' the same groups, etc. Any exceptions, and you have broken the isolation between the accounts.