Train to/from Amsterdam -Paris

The main provider of rail service from Paris to Amsterdam is a Thalys high-speed train for which the ticket prices do change dynamically, like on airlines (and on Eurostar and France's domestic high-speed/TGV trains), so if you wait until the day of, you risk sold-out trains or high prices -- a look at Paris-Amsterdam trains tomorrow (May 31) has exactly 1 train at €97 with the other ones at €135, with a couple of trains that are completely sold out.

If you already know your travel dates and are comfortable with non-changeable/non-flexible tickets (usually the early-purchase, cheap tickets cannot be changed/refunded or can only be changed for a fee) then you should buy the tickets now.

There are apparently also now other providers of rail service on this route, but I can't speak to any of the other ones as I am not familiar with them.

You can buy tickets at the station on the day, same as in Italy, and many people travelling at short notice (mostly for work) do so.
The advantage of buying weeks in advance, is it can cost 1/2 to 1/3 of the price you pay on the day. The disadvantage is that a ticket is for a specific train (it says date, train and seat numbers on it) and cannot easily be changed.

You can buy tickets at the rail company website: https://www.thalys.com/ This will get you a print-at-home ticket, which is all you need to get on the train.

Just arrived in Paris using Thalys from Amsterdam on Monday. I paid 40€ some weeks ago for tickets. Tickets were available the day we left at 135€. The train was absolutely full and trains being sold out are not uncommon.

I would purchase tickets as soon as you can.

first time I took the Thalys many years ago I paid over 100 Euro the week of travel; since then I have always booked well in advance and paid 35 Euro. That is a pretty giant savings. It is similarly true of other high speed distance travel -- buying months out gives you the best prices.

Italy also has advance-purchase ticket discounts, but you very likely were taking those trains for shorter and/or cheaper distances than this one, making the advance savings less valuable on that earlier trip.

While there are advance purchase discounts for domestic trains in Italy, the savings are not as drastic as they are for the Thalys. Full price tickets in Italy aren't that expensive; full price Thalys tickets are (at least for my budget).

Particularly since you know your travel dates, there is no advantage to waiting, so book now for best prices and selection of trains.