The floats are a surprisingly bad approximation of the reals. They're barely better than the integers, while their shortcomings are much harder to understand, so I'd say it's like if you want approximately a spaceship and you're choosing between an F-14 Tomcat and a Fiat Punto.
Neither of these things is a spaceship, but, it will be obvious to almost anybody why the Punto isn't a spaceship whereas you are less likely to know enough about the F-14 to see why it's not "good enough".
I think the truly surprising thing is just how well floating point numbers work in many practical applications despite how different they are from the real numbers. One could call it the "unreasonable effectiveness of floating point mathematics".
There are many situations where you have something you want to compute to within low number of units in the last place, that seem fairly involved, but there are very often clever methods that let you do it without having to go to extended or arbitrary precision. Maybe not that surprising, but it's something I find interesting.
x = 0.5 (with bit pattern 0x3f000000)
y = 0.25 (with bit pattern 0x3e800000)
then
x+y = 0.75 (with bit pattern 0x3f400000)
But if we just added the bit patterns like integers, we would get 0x7d800000 (with a float value of over 10^37). Just because they are discrete doesn't mean they are integers only that they can be mapped one-to-one with integers. The bit pattern is not the semantic meaning of the value, and you cannot perform correct operations if you ignore the semantics.
Neither of these things is a spaceship, but, it will be obvious to almost anybody why the Punto isn't a spaceship whereas you are less likely to know enough about the F-14 to see why it's not "good enough".