The CPU microarchitecture has changed a lot from 2008 to present day. If we think only about Intel models, the memory controller and the GPU were incorporated to the CPU, Core microarchitecture advanced six generations, and the manufacturing process was reduced from 45 nm to 14 nm. With this, a basic/mainstream CPU like the Core i3-4150 can be significantly faster than the Core 2 Quad Q8300, which was a high-end processor, even with only two physical cores, against the four cores of the Core 2 Quad Q8300.
However, real low-end processors like the Pentium N3700 and the Athlon 5150, even also with four cores, did not reach the processing performance level of the Core 2 Quad Q8300. Of course, they are based on low-power microarchitectures, not focusing on performance. And, to be honest, for most current situations, it is better to use a CPU that dissipates only 6 W like the Pentium N3700 than a Core 2 Quad Q8300, which has a performance between 4% and 81% higher, but uses 1500% more electric power.
Another important subject is about the integrated video. On the Core 2 Quad Q8300 (as well as on all LGA775 CPUs), the onboard video is controlled by the chipset, not by the CPU. We could notice that the nowadays integrated video, even on the most basic CPUs, is faster than the solution provided by the G41 chipset we used with the Q8300. So, we can conclude that we are less dependant on a dedicated video card now than we were some years ago; today, you need a video card only on a “gamer” PC, or on very specific applications that need high video performance, or if you use a software that explicitly takes benefit on GPGPU technology.
So, our original question does not have a single answer: an old CPU can, in certain cases, be more powerful than a modern one. But, depending on how old is your CPU, even if it was a high-end model at the time, it can be outperformed by a mainstream or even by a low-end modern processor.