There have been some posts about the danger of buying a business as an ongoing concern, and continuing trading as before, as there may be skeletons in the closest, which makes you the new owner now the responsible person.
There is also a danger for the seller, as there may be sureties in place, which suppliers do not wish to annul. The new owner can then trade recklessly, and you as the seller become responsible for the bills, as you are still standing as the original surety holder.

There is discussions and agreements between suppliers and customers when doing this, but I have found that in the majority of cases, none actually mind, provided that the business continues buying/supplying as before.

There are ways of maintaining the same/similar name but a new entity, but this is for another day.