Ok, I was half-right.....sort of.
Your router supports the 5GHz 802.11n standard - 5GHz is a less congested signal & has the ability for faster throughput.
I would assume your granfathers rig is connecting to the 5GHz signal (assuming it's also wifi). - Can you confirm the model of the adapter in his setup? (Device Manager should help)
Your wifi adapter is limited to the 2.4GHz frequency - meaning you're on the most congested of the two (*most* cordless home phones, bluetooth devices, even microwaves all use the 2.4GHz frequency) - there is a bit of a speed difference too, but the congestion is likely the main culprit.
I found this:
http://forums.timewarnercable.com/t5/Connectivity/TG1672G-Modem-doesn-t-support-802-11ac-WiFi-for-300Mbps-Speed/td-p/106185
They also mention a 300Mb/s plan, so it must exist.
So, you have some options. The simplest, most stable option would be to run an ethernet cable to your setup, but I'll stick to the wireless option.
Option 1 - speeds similar to your grandfathers
Upgrade your wifi adapter to one that can utilize a 5GHz signal. Most AC wifi adapters can utilize wireless N on a 5GHz signal - not all can though. There will be some wireless N options that can use 5GHz, but a lot (like your current one) are limited to 2.4GHz.
Option 2 - Upgrade speeds dramatically
1*. A modem/router upgrade, to one supporting 802.11ac - ideally TWC would be able to provide this, but the forum post is still pretty new, so I assume they can't/wont'. Your grandfather can still connect his N device to
2*. Purchase an 802.11ac repeater/access point, run an ethernet cable from your existing router to it, and configure for use.
*Either way, you will need a wifi adapter upgrade (the one I linked should be perfect, but if you want PCIe options, I can link some too).
I have a 100Mb/s plan that I exceed slightly (see my sig) with the adapter I linked in my first reply.
Realistically, without either a router/wifi adapter upgrade (or at least an AC access point & wifi adapter upgrade) a 300Mb/s plan is pointless. You'll never achieve speeds remotely close to those on your existing hardware - and I find it ridiculous that's the modem/router that TWC are providing on the plan.