To me, the former suggests a lottery winning or inheritance while the latter suggests discipline.
From the point of view of credit scoring models (in the US), once all the debt is paid off, those two behaviors are equivalent. The models are designed to predict if you will pay off potential new debts, not how you will do it.
As for how your score moves, it depends mostly on where you started before the payoff. If your score was very high, it will probably drop a little, and if it was very low, it will probably climb quite a bit. In general, there are two major changes that will occur:
- Your revolving lines of credit (credit cards and lines of credit) will go to nearly 0% utilization (unless you have unusually low limits and continue to use the cards for day to day). If your previous utilization was high and now it's nearly 0%, this will significantly increase your credit score.
- When you pay off all of your term loans such as auto, home, student loan, personal loans, etc, your score goes down a little bit. This is because your credit mix is reduced.
It's important to note that the above is true regardless of whether you pay off the debt all at once or spread it out over time. If you are comparing doing it now versus spreading it out over the next 12 months, then 12 months from now, your score will be basically the same either way. I say "basically the same" because the reality is with the slow method over 12 months, your term loan accounts will build up 12 more months of history so your average age of accounts will be slightly higher, meaning your score might be a little better off as a result of the slow method. However, that difference should be minimal unless your overall credit history is short (say less than 3 years).