Does it matter which specific assets I list versus just buying 'the business' generally?

Yes, and this is one of the more consequential drafting details in an asset purchase. An asset purchase agreement works by listing exactly what's included — equipment, inventory, contracts, intellectual property, goodwill — rather than transferring "the business" as a vague whole. Anything not listed generally isn't included, and anything not expressly assumed as a liability generally stays with the seller.

The trap is assuming a general description like "the business and its assets" covers everything you actually need to operate it. If a specific piece of equipment, a key contract, or a licence isn't itemized, you may close the deal only to discover you don't actually own something essential to running what you thought you bought. This also matters for tax: a joint election that can apply so GST/HST doesn't apply to the sale generally requires that the buyer acquire all or substantially all of the property needed to carry on the business — a vaguely described deal makes that harder to support.

Working from a detailed schedule of assets and assumed liabilities, built with a business lawyer rather than relying on general language, is what actually protects both sides here.