Is a 2.4 GHz Intel Core 2 Duo better than a 2.0GHz dual-core Intel Core i5 with turbo Boost up to 3.1GHz?
Don't be fooled by the Ghz. The i5-6360U is in the Late 2016 13" MacBook Pro and is a Skylake CPU. I assume your 2008 MacBook Pro is a 15" model, as the 13" did not ship with that CPU.

The newer i5 machine has a Geekbench score of 3609 vs 1320 for the Late 2008 MacBook Pro 15". 

Not only that, but the newer Mac features better graphics (even integrated graphics), built in SSD, etc. If  you still have a hard drive in your old Mac, the SSD alone will shock you in performance improvements for your Mac.

GeekBench

Update: I see some additional questions in the comments about how a 2 Ghz processor can 'beat' a 2.4 Ghz processor. One thing to note is that the transition at Intel from Pentium to Core Duo essentially ended the Ghz wars. Intel found the adding cores, reducing heat and improving cacheing improved performance more than Ghz. This really showed when their mobile CPUs were beating out their desktop CPUs, so Intel switched to their 'mobile architecture' for everything, thus Core and Core Duo.  The i Series are built on that philosophy. 

Another way to look at it is this: Most common car (US) V8 engines in the  70s produced as little as 110 HP. (Muscle cars made much more, but enlarged the engine and got terrible gas mileage) Today, we have 4 cylinder engines that produce 300 HP, and V8s that produce muscle car HP but get economy car fuel mileage. So, how does a 4 cylinder make more horsepower than a V8?  Its far more than the number of cylinders, just as its is far more than Ghz for CPUs.