What is the advantage(s) to buying tickets beforehand vs at the station for rail travel in Europe?

If you already know at what dates you'll be traveling, then yes, there is an advantage. All trains between London and the continent, all decent trains between Paris and Brussels, and all convenient trains between Brussels and Amsterdam are high-speed trains with compulsory bookings. The more in advance you book, the less risk there is that the train will be full, and the less expensive the fare will be.

You can get a print-at-home ticket-and-reservation for all of these journeys over the Internet. Shop around: sometimes international fares can be different if you buy them from one country or the other or from the train operator. Print-at-home tickets are nominative, you can't resell them to someone else and you'll need to show your passport or other identity document to the ticket collector.

For shorter journeys on regional trains, there is no reduced fare and no reservation, so there is no advantage to buying a ticket in advance except not having to queue at the station. Beware that in France, train ticket selling machine require a credit card with a chip or coins; there is no surcharge for buying a ticket at a window but there can be a queue. In the Netherlands, ticket machines require a Dutch banking card or coins and there is a surcharge for using a ticket window.

Besides the compulsory booking aspect, a very important factor may be the fare. Check for instance TGV trains between Paris and Brussels. The changes for fares for different dates are considerable.

If you're planning on taking the Eurostar from London, it's important to book ahead of time since (a) you can often get a better deal if you watch the prices and pick out the cheapest time/date that fits in your schedule, and (b) these trains fill up quickly so booking a ticket ahead of time ensures that you have a spot on the train that you want. See eurostar.com. For other trains it's not as important, but you'll need to arrive earlier at the station so you can purchase your ticket. Purchasing ahead of time is often cheaper and easier, if you know your dates. The important thing is noting that there are two types of trains - those that require a reservation (high-speed trains, overnight trains, and trains between London and continental Europe) and local trains that don't require a reservation; local trains usually don't fill up as quickly. I'd definitely advise you to book trains that require a reservation ahead of time whenever possible.