Why do float and int have such different maximum values even though they’re the same number of bits?

Your intuition quite rightly tells you that there can be no more information content in one than the other, because they both have 32 bits. But that doesn’t mean we can’t use those bits to represent different values.

Suppose I invent two new datatypes, uint4 and foo4. uint4 uses 4 bits to represent an integer, in the standard binary representation, so we have

bits   value
0000       0
0001       1
0010       2
...
1111      15

But foo4 uses 4 bits to represent these values:

bits   value
0000       0
0001      42
0010     -97
0011       1
...
1110      pi
1111       e

Now foo4 has a much wider range of values than uint4, despite having the same number of bits! How? Because there are some uint4 values that can’t be represented by foo4, so those ‘slots’ in the bit mapping are available for other values.

It is the same for int and float – they can both store values from a set of 2³² values, just different sets of 2³² values.