They're related but they're not the same deal. Two words, two structures.
Subject-to means the deed transfers to you and the seller's existing loan stays in his name, at his rate, and you make those payments. There's no new loan created between you and him. If you pay him for his equity over time, that's a separate small note, and it sits alongside the deal rather than replacing anything.
A wrap, short for wraparound, also leaves the underlying loan in place, but the seller creates a brand new note to you that wraps around it. You pay the seller on the new note. The seller pays his lender out of what you send him. Sellers like wraps because they can write the new note at a higher rate than the underlying and keep the difference. So on your numbers, he could wrap 280k at 6% while his own 3.25% loan keeps running underneath, and pocket the spread every month.
The loose usage you ran into is real. Plenty of people call any deal where the old loan stays in place a "sub2." When someone says that word, ask whether a new note is being created and who mails the payment to the lender. Those two answers tell you which structure you're actually in.
Cash-wise, the 40k is the seller's equity, and it's negotiable, financeable, or sometimes much smaller when a seller is behind on payments. You'll also fund reinstating any arrears, closing and attorney costs, and the escrow the lender holds. Recording and disclosure rules for these transfers vary by state, so have a real estate attorney in that state paper it.