I'm located in Belgium as well, my accountant has no issue with me not sending them invoices. I think he books it as a commission.
Since your "customer" (Google) is located outside Belgium (Ireland), you do not have to collect VAT from them (neither do you have to give it to our government).
VAT between businesses is supposed to be a nul operation: you collect it from your customers on behalf of the government (when appropriate) and give it to the government - it's never yours... The exceptions (numerous as they are in a country with the 4th highest taxes in the world) are all related to how you can deduct VAT you paid to your suppliers from the VAT you have to pass on to the government.
All I do is print the monthly reports (showing the day by day estimated revenue) as well as the monthly statement from Google (under Payment history > previous payments > details > statement of earnings), and give that to our accountant.
The details are enough to justify it all. The statement of earnings gives the details on who your customer is, the amount etc - including the IE VAT ID-. It lacks the "details" required on invoices out here, but the report covers that in more detail that they'll care to review.
There's also once their reference to the EU directive that's printed and stored should the Belgian government get worried about that (they won't they know Google will just recuperate it should they force us to charge it).
Does this mean there's no practical difference between an individual and a business account,
There is a big difference: if you collect the money in a business you're safe and legal. If you collect it privately, you need to do a lot more to make it legal: it's now undeclared income till you declare it (and hence pay income taxes on it) - and both Google as well as your bank might well be required by law to blow the whistle on you.
Undeclared income might cost you a lot if you're not self-employed: if you have too much of it they hit you a dozen times with fines (first a fine for not being self employed while you should be, then one for not declaring income as self-employed, then one for not paying taxes being self employed, for not paying social security, ... and on and on. It's risky to not be self-employed (at least after hours) and have income that you cannot predict how much it'll be in a year.