It is important to understand the difference between assuming a mortgage and being subject to the mortgage. When a buyer buys property and assumes a mortgage, the buyer becomes primarily liable for the debt and the seller becomes secondarily liable for the debt.
"Assume" means the buyer takes on liability, and the seller is no longer primarily liable. "Subject to" means the seller is not released from responsibility.
The word "assumption" is used when a buyer assumes personal liability for an existing debt. If the buyer defaults, the seller no longer has responsibility as the buyer has "assumed" the loan.
The term "taking subject to" is when the buyer incurs no liability to repay the loan. The loan stays in the seller's name, but the buyer gets the deed and therefore controls the property. Although the buyer makes the mortgage payments, the seller remains responsible for the loan. When the property is sold subject to the loan the buyer is not liable to pay the lender, the original borrower is still primarily liable to the lender.
A qualified veteran negotiates a VA guaranteed loan to purchase a home. He moves to another state. The new buyer purchases the property subject to the mortgage. Because the new buyer is not a qualified veteran, if they default on the loan the veteran is still primarily liable to the lender.
When a buyer "assumes" a loan, it has to be done with lender's approval. An assumption agreement is prepared by the lender and signed by the buyer during escrow. The lender typically would require a credit history from the buyer before approving the assumption and the payment of assumption fees, the same way as if the buyer was qualifying for a loan with that lender.
When a buyer takes title to property "subject to" the loan, the lender is not notified of the transaction or asked for their approval, because the seller is not released from responsibility. The buyer is making the payments instead of the seller in this case. The seller will be asked to provide escrow with their last payment record, which will be used to calculate the exact principal balance at close of escrow. The lender may have the right to accelerate the loan when/if they get notified of a ownership transfer.