The difference is purely in the maximum value which can be stored (18,446,744,073,709,551,615 for the bigint(20) and 4,294,967,295 for the int(10), I believe), as per the details on the MySQL Numeric Types manual page.
Incidentally, the use of (20) and (10) is largely irrelevant unless you’re using ZEROFILL. (i.e.: It doesn’t actually change the size of the number stored – that’s all down to the type.)
However, in practical terms it should be noted that you’re not likely to hit either of these limits any time soon, unless you’re a really active blogger.