The major advantage trains have to belts is simply the fact that you can have multiple trains use one set of tracks to go to different destinations, Though you need to worry about collisions in that manner unless you build a more complicated system with 2 lanes.
If you build one advanced network of train rails, you can then have dozens of trains use the same rails to get to different destinations. Set up one Big copper mine that makes copper plates, and then you can build a train stop anywhere connected to the network, and have a train haul only copper plates from that original stop to your new stop. The copper will get from across the map from A to B, and it will only be copper. while you only had to build a few rails to do so since up to 90% of it will have already been done for hauling other resources. On one single track, you can have any and all resources travel over it in your trains, but will never mix and always get to where you need them in a timely fashion.
This of course is very difficult to accomplish, as it is very complex. The biggest Let's players work around this by doing it modularly, so that all of the hard work is only done once and they have drones build all but the fine details. This allows them to find a place with the raw resources they need, and then can bring over whatever they need to finish production. Building Belts for one or two lanes is certainly more efficient than setting up 1-2 trains, but if you need to get lots of resources to lots of places, then your going to need dozens of lanes of belts that are likely going to need to cross over each other and are only good for one thing each, But 2 train rails with a few stops can do the exact same thing and more.
Don't forget each cargo wagon can hold 40 stacks of items, which is 2000-4000 items for most resources. Each belt can only hold 6-8 items (ignoring throughput here), which means an average train with 3 cargo wagons can hold more resources than 1000 belts, but only take up the space of about 50 transport belts.
And there is always the ability to ride the trains for high speed travel.
Edit: Something else to consider is storage at the train stations themselves. The most popular thing to do is to set up chests and have inserters go from chest to cargo wagon directly for the stack inserter size bonus. I don't recall the exact number, but i believe there are 6 slots on each side of the wagon, for up to 12 chests of storage, both on the loading and unloading Train stops, for an additional 24 chests (which could be steel chests) of storage of resources per cargo wagon, for 72 chests for a 3 wagon train. This is a massive buffer of resources that could be filled while demand is low, and then last a long time when it goes up.
If someone knows the math, feel free to replace this with it, but a train system with at least moderately upgraded Inserters and 2-3 wagons could likely complete if not beat red belts in terms of throughput thanks to their capacity and speed. Of course you could always just add a 2nd or 3rd train, then it probably could blow blue belts out of the water.