The biggest difference is that an SPA is the sale of all shares, and an APA is the sale of selected assets. Therefore, they are both different transactions and have different procedures.
First and foremost, it is typically the buyer's responsibility — not yours as the seller — to draft the Definitive Agreement. This will not begin until both the buyer and the seller sign a Letter of Intent indicating their intention to buy/sell the business.
Asset sales offer tax advantages and selective asset acquisition, but can be complex and require additional time and costs. Equity sales provide simplicity and continuity, but require the buyer to assume all liabilities. Both types of transactions involve important accounting considerations and post-close diligence.