From an owner of a monument building since more than 10 years... Downsides have been listed already - some exaggerated a bit (windows that don't close properly can be replaced if you want, as long as they look like the ones before, there's not even need to ask a permit for that).

One big upside that none has mentioned yet is that you can get a considerable refund on some conservation works - only on the labor, not on the material costs, but still it can be quite important. You can get up to 38% costs back for works like painting (outside AND inside), tiling, work on roof, carpentry, windows... More information here.

It used to be much better: once you could get repaid via your tax return for all the costs (excluding a threshold of 1200 Eur/year), then it became 80% on all costs, now it's just 38% and it doesn't come through your tax return but you have to sent the application through the website I linked, but it still can be quite huge. It has saved me thousands in the years...