Can you get a discount by booking trains early, as in Italy? Or is the fare going to be about the same no matter when you buy your ticket?

Yes, there are advance purchase discounts.

There are some very good discounts available for booking early, but you have to wait until the schedule is posted. There are also discounts when making a round trip purchase.

You can easily find the answer to your advance price question in concrete manner by looking at http://www.renfe.es and compare the last minutes prices, March 16 and furthest advance tickets available today = June 11.

For exmaple, Sevilla-Madrid

March 16: AVE 2151, €127.20 Preferente
June 11: AVE 2151, €34.75 Promo

Are these about the same no matter when you buy your ticket?

Some trains are worse than expensive. No seats left such as

June 11:
16:45 €40.75
17:45 €40.75

But

March 16:
16:45 train full
17:45 train full

We bought tickets easily through petrabax.com (as noted by seat61.com) and found the official Renfe site to be balky at best. There is a small booking fee for using Petrabax.

I had no trouble using renfe.es. I also discovered that it is considerably cheaper to buy on the internet than in person at a train station.

Greg, is there generally a set number of days that Renfe starts selling advanced (sale) tickets for a given date? In doing some research a couple of weeks ago for a trip we're planning for next March, I read conflicting things about when tickets for any given date go on sale.