Why didn't the Phenom I system have the problem then?
Driver issues? Malware? Faulty? Iffy browser extension? Did you reinstall Windows when you changed motherboard? If so, is it set up the same (and if not it could be a clash of chipset drivers)? Is your Windows Power options set to "balanced" and not "power saver" (which would lock it in at its idle state). Some things you describe are normal (slower in some heavy games vs quads), but others "massive slowdowns when downloading", "web pages grind to a halt" aren't, even for dual cores (if that were "normal" then most 2-core 1.0-2.5GHz laptops wouldn't work even for simple tasks - clearly that's not the case...)
As a "reality check" to what modern Pentium's are capable of - If I disable 2 cores (in the BIOS) on my i5-3570, then lock it in at 1.6GHz mimicking a really slow mobile Pentium half the speed of a 3.0GHz Haswell G3220, even at 2x 1.6GHz cores, with a background MP3 in Winamp + playing a 1080p Youtube video in Firefox + HTPC software background recording from TV tuner card + reloading Anandtech's home-page is virtually instant. Average CPU usage for all 4 tasks combined is 26% / max is 45-50% (of 1.6GHz = 416-800MHz used per core).
Here's a screenshot if you want proof.
When clocked at desktop 2x 3GHz cores, average CPU usage falls to 15%. With 4x 3GHz cores, it drops to below 8%. But in none of these is any CPU core remotely close to being a bottleneck even when doing 3-4 things at once, even when running at half the speed of your CPU. Even if you run a badly written game that loads 1x core 100% when minimized, the remaining core should still be enough for basic web browsing.
Likewise simply "having web browser tabs open" uses memory more than CPU (and you can speed up any system far more with adblock). Loading a web page with browser cache set to a 5,400rpm laptop drive gets massively HDD bottlenecked (lots of random <4k file read/writes which is "worst case" for HDD's - down to 0.5MB/s transfer rates when accessing dozens of tiny 4k size files (ie, typical web cache cookies, css/js files, etc) simultaneously on a 5,400rpm drive). Sometimes it can even be slower than disabling HDD web cache. Is your HDD LED constantly on when these browsing slowdowns occur? As mentioned, try taking a screenshot of CPU usage / using a CPU usage logger when browsing the web. Long install times / unresponsive web browser when installing large programs to HDD, are also due to a HDD bottleneck. Even high-end quad/octo cores run like treacle if they try to heavily access a 5,400rpm HDD which is being 100% loaded by another process. SSD's really are worth the hype & money on any machine down to and including netbooks.
When opening new web pages the cpu spikes to 30-40% while the HDD spikes to 70-100%
With the web browser open looking at youtube(just the main page) or anything for that matter the HDD spikes from 0-100% every 3-5 seconds.
Edit : Just noticed above. That is absolutely a HDD not CPU bottleneck.