There are two methods of acquiring an interest in a business that is being sold as a going concern.
One is to buy shares of the company that owns the business (i.e. the company owns all the assets and liabilities which make up the business). In this case the company owns the business both before and after you acquire some or all of the shares issued by that company.
The other is to identify and buy specific assets (and perhaps take on specific liabilities) which together make up the business you want to acquire. In this case a different legal entity, perhaps a company set up by you, will become the new owner of the assets and liabilities which make up the business.
The key legal difference is that in buying shares in a company, the value of your shares can be adversely affected by any liability of the company whether or not you are aware of that liability when you buy the shares.
Conversely if you buy from a seller a business as a going concern (i.e. a list of assets, specific liabilities, and employees) all other assets and liabilities of the business, and its anything arising from its tax and trading history, remain with the seller.