If we're going to do comparisons, allow me compare to a system that I'm very familiar with and have logged about 50,000 km on: Russian Railways. I think it's a fair comparison because both Russia and the U.S. are large countries, and both have long-distance trains that travel great distances. So, let's begin:
FREQUENCY OF SERVICE: Russian railways has much better frequency of service. A wide variety of passenger trains (more on that later) traverses mainlines every 1-2 hours. You won't have to wait long to catch a train between, say, Novosibirsk and Krasnoyarsk. Off mainlines, service is less frequent, but generally is at least several times a day.
SLEEPER ACCOMODATIONS: Russian long-distance trains are composed of sleeper cars only. There is no coach seating: Russians shudder at the thought of spending the night in a seat. There are 3 classes of sleeper cars: SV (2-person compartment, 1st class), kupe (4-person compartment, 2nd class), and platskart (open configuration, 3rd class). A person traveling alone has a choice of buying out the whole compartment or being assigned roommates. Gender segregation is optional, but women traveling alone generally request female roommates. In addition, there are 3 classes of trains: firmennyi ("brand-name", top-of-the-line), skoryi ("express", middle-of-the-road), and passazhirskiy ("passenger", slow train, makes a lot of stops). This provides a lot of flexibility: a person wishing to be pampered can book SV (first class) on a firmennyi train (here's a random example:
http://visual.rzd.ru/isvp/public/visual?STRUCTURE_ID=1039). On the other end of the scale, the cheapest way to travel is platskart (3rd class) on a passazhirskiy train: you are in a large dorm room on wheels, there are often issues with BO and other unpleasant smells, there is typically no air conditioning, and the toilets can be downright nasty. Amtrak sleeper service is about equivant to that on a skoryi (middle-of-the-road) train: the attendants there will provide the basics, but will not bend over backwards for you. One positive thing about Amtrak, though, is that all long-distance trains are equipped with showers. In Russia, generally only firmennyi trains traveling more than 2 days have shower compartments, and even that is a relatively new addition.
ON-TIME PERFORMANCE AND RELIABILITY: Generally a lot better in Russia. As in the U.S., passenger trains share tracks with freights almost everywhere, but passenger trains are always given the highest priority. A long-distance train that is delayed by more than an hour is considered "extremely late", and it hardly ever happens. Cancelled trains and bustitution is pretty much unheard of. Mechanical problems are rare.
OVERALL IMPRESSION: Traveling by train in Russia has generally been a better experience for me than in the U.S. (I am fluent in Russian, incidentally, so language hasn't been a problem). There is frequency, flexibility, consistency, and reliability that is lacking in the U.S. First class on long-distance trains is much nicer than Amtrak, third class is generally worse, but you get what you pay for. There's train service to just about all major cities, save for one or two in the remote reaches of Siberia (Yakutsk comes to mind, but a train line to it is being constructed right now). So whenever someone argues that trains will never work for the U.S. because it's "too big" or the population density is "too low", I always cite Russia as a working example.