I'm going to guess that YouTube is insisting on using VP9 for playing videos on anything that isn't Internet Explorer. VP9 doesn't have widespread hardware acceleration support, so pretty much every single VP9 video has to be decoded using the CPU. The Pentium E5200 is likely too slow for the task without any assistance from the GPU, so that might explain why Internet Explorer plays the videos smoothly - Internet Explorer supports only H.264, for better or worse.
Compare these two Pentiums to see how much the baseline performance has increased in the years in between. Consider either changing YouTube so that you're opted out of VP9 codec, use an extension to automatically prevent YouTube from playing videos using the VP9 codec, or get a new system, since Pentium-AE boxes give you a ridiculous amount of performance compared to what you have right now while still being as stingy as you can in terms of budget, even without overclocking.
(Really, on a CPU that weak, you really should stick with Internet Explorer 11 and/or Microsoft Edge.)
CPU + motherboard upgrade should give you a massive boost in terms of CPU and GPU performance. Intel HD Graphics is a whole lot better than the old motherboard graphics, and the CPU is, well, awesome enough on its own. (Get a Z97 motherboard if you really want to overclock, but I think most H97 has unofficial OC support.)
