If you are filing for Chapter 7 bankruptcy, there is a chance that some of your possessions may need to be sold in order to repay creditors. Your letter intent should describe what assets of yours will be kept and which will be sold, as creditors will want to be informed of this information. These letters should be sent within 30 days of filing for bankruptcy. A lawyer is typically used to ensure that these letters get to all the right places and contain the proper, and correct, information.
Note that the definitions are not too far apart, but the documents themselves are. The biggest difference is a Purchase Offer is a formal legal document, that once signed by both parties becomes a legal and binding agreement. If you are working with a real estate agent or broker, this is usually the document they will have you sign if you are interested in placing an offer on a property.