You have to know bottleneck. At all places only bottlenecks affect performance. In the case of Internet what's the bottleneck? - there is CPU speed, RAM, the ethernet card on your device, speed of your wifi router and then the speed of your Internet connection.
Most times router speed is so less that other matters wont affect. Most of the ethernet cards now support 1Gbps and internet speeds are much lower. So, that problem goes out. Similarly, most wifi routers supports at least 54 Mbps and unless its a bad one, at least 5 people could use the same without any performance problem. Now, most of the websites consume very less CPU/RAM and hence speed of your internet connection is the only bottleneck. But in case of playing online games or loading pages with high flash content, javascript uses the CPU of your device. So, these sites will load better in a better system.
Now, just the CPU speed is not the criteria for a better system. A low CPU speed system may have more cores. Each year many better architectures come which have increased caches and more bandwidth among system components. These are at least as much important as CPU speed. Hence, a newer architecture processor with lower speed may perform far better than an older architecture chip with more speed.
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