Well, the Social Security numbers will be linked, so that Social Security and the IRS can coordinate information stored under the different Social Security numbers.
As you are aware, there are many people to whom you have given your Social Security number (bank accounts, credit cards, insurance, possibly medical offices and utility companies). You will either have to drop your contacts with those entities, or tell them about the new Social Security number. If you don't contact your creditors, they will still be able to "hound you" under the old Social Security number. (It's up to the court if you can legally deny being the person under the old name; this is certainly not within the field of "Social Security" questions, so I cannot even speculate as to what you are legally allowed to say.) If you do contact your creditors, they will be able to link the old and new numbers; whether they do so and report the link to the credit agencies is up to them.
If there are no links to your new Social Security number, you will have a blank credit record with that number, which is not exactly the same as a clean credit record. This isn't the witness protection program, where the government will create a new identity with some history.
Again, you asked the question in the "Social Security" category; many of your followup questions fall into different categories, where I may not be an expert.