When you buy a "going concern," you purchase the entire business, including all of its assets and liabilities, and continue to operate it as it was operated before the sale. The only real change is the change in ownership. You get the benefit of the business's goodwill, or reputation; you are obligated under its contracts; you are entitled to its receivables. Normally, you acquire the business entity, either through the purchase of all of its units of membership in the case of a limited liability company, or all of its shares of stock in the case of a corporation. Because you are taking on the liabilities and contractual obligations of the business, it is very important to do due diligence and to have a carefully drafted purchase contract.
In an asset purchase, the selling entity remains intact, but you acquire some or all of its assets. You may add these assets to a business that you already own, or you may want to form a new business entity to hold them. Generally, you are not liable for the selling entity's liabilities, but there are some exceptions. For example, the Wisconsin Department of Revenue can assess successor liability for certain types of tax owed by the seller, such as sales tax, against the buyer, even in an asset purchase. Additionally, the assets could be subject to liens.