If you are running simple word press or blogs, you can never see the changes so that you do not need to spend more money to get a dedicated server. but if you are running a website which has many hits or running high frameworks/themes, we recommend dedicated server, because it give all resources to your requirements only. The resources/specs of dedicated server will be depends upon requirements of your website.
If you have vps from any VPS provider, they sell VPS in the same node to other customers also, As our knowledge everybody showing vmem, vcpu and network which all are sharing with other customers also. It provide resources to the VPS which requests resources. We never see that anyone provides dedicated CPU for vps.
You can check the performance of dedicated server by checking the network speed(download/upload speed), HDD/SSD write or read speed. Memory can test by memory test which is available in all Linux or windows ISO and may tools available to check processors. Also check the RAM is latest DDR4 ECC and its frequency.