Let's make it clear from the beginning (if it isn't too obvious already) that I love old riveted steel bridges. Watching these old rusty skeletons on a sunny day where the brown colors really glow is something special indeed. As a bonus, often the foundations for these bridges are made of superb stonework where each individual piece is so precisely aligned to its neighboring stones that not even a thin sheet of paper fits in between.
Where most bridges today are reinforced all-concrete structures, it actually took quite a while before the construction engineers learnt how to make them tough and durable enough to withstand the dynamic loads of traffic and heavy trains. Although there are some early concrete bridges built before 1930, it was not until it was understood that the pre-stressed concrete was the way to go to. With introduction of computer modeling and simulations in the late 1960s, bridges became lighter and stronger and today’s concrete bridges are actually quite aesthetical.
With increased traffic, higher speeds and heavier axle loading, existing steel bridges (and even early concrete ones) have typically been reinforced and/or widened over the years, but at a certain point it is either not economical or practical to go any further. To keep traffic uninterrupted, usually a new bridge is built beside the existing one and when the new one is finished, a quick swap is made. When the new bridge is in full operation, the old one is blasted and usually the only remains (if any) are the foundations at both ends.
In some cases however the old bridges have been left as is. I assume that although there may in some case be some strategic thought of keeping a backup, the most common case is that demolishing the old bridge and taking away the remains is so costly that not even the value of the scrap offsets it.