However, since both wired and wireless connections are so fast nowadays, in your day to day use of the internet its very possible that you will not notice a difference at all between wired and wireless. For this reason, I use wireless on my laptop at all times just to avoid the minor hassle of plugging in the ethernet cable.
You're mixing two completely different things here. OP is asking about connecting computers
within his LAN, and you're mentioning accessing internet. It's also incorrect that wired and wireless are "so fast", in fact wired always had a big speed advantage over wireless, difference that nowadays is still here (just search for specifications of both).
Switching from wireless to wired will, therefore, have a great improvement in network performance
within his LAN. This will be evidenced, as you said, in data transfer between both computers being much faster.
Another completely different topic that you seem to be mixing is internet access. Download/upload speeds depend mainly on the plan contracted with the ISP much more than the LAN speed. Like in all networks, speed is based on the slowest fragment, and the bottleneck is almost always the ISP rather than the local connections. It's also not what the OP asked.
So, while it's true that the switch is unlikely to have an effect in internet speed, it's not due to wired being similar to wireless, but due to the weakest link in the chain, the ISP, remaining unchanged.