From a theoretical perspective, there is a slight increase in entropy because the position of the break between the first and second passwords constitutes additional information.
One big password. Let's say you have one password and the user enters 16 characters, and for simplicity let's say that the password must be numeric. This amounts to 10^16 possible permutations, or 10,000,000,000,000,000 (10 quadrillion).
Two passwords. Now let's say you have two passwords that total 16 characters. You would think there are 10^16 combinations, but in fact there more. Let's add them up:
If the first password has 1 digit and the second has 15 digits, that amounts to 1^10 x 15^10 = 10,000,000,000,000,000 permutations.
If the first password has 2 digits and the second has 14 digits, that amounts to 2^10 x 14^10 = 10,000,000,000,000,000 permutations.
If the first password has 3 digits and the second has 13 digits, that amounts to 3^10 x 13^10 = 10,000,000,000,000,000 permutations.
If the first password has 15 digits and the second has 1 digit, that amounts to 15^10 x 1^10 = 10,000,000,000,000,000 permutations.
There are 15 possible cases, totaling 150,000,000,000,000,000 permutations (150 quadrillion).
Comparison. Since 150 quadrillion > 10 quadrillion, yes, there is a bit of increase in entropy and therefore it is harder to guess the two passwords than the one long password.
The assumptions above (length = 16, character set = 10) can be changed and the math will still work out.
On the other hand, depending on how the system evaluates the password, the above may not apply, e.g. if the web site combines the passwords and hashes them together instead of keeping them separate. If that is the case, then no, there is no additional computational power required to forge the password, because there are in fact 15 different password combinations that will work. 150 quadrillion ÷ 15 = 10 quadrillion so you're back where you started.
It's even worse!
See lengyelg's answer which is spot on:
It's actually less secure to have two separate password fields in the sense that if password hashes are stored separately, it can be easier to find two shorter passwords from something like a rainbow table than one long password. Of course if a single hash is stored for the concatenated password, it's the same as one password field.
In my opinion, it is far simpler to require an extra digit than it is to require two separate passwords, and that will always have a stronger effect if your character set is larger than the length of the two passwords combined (which will almost certainly be the case).