If we consider the difference in tech between the old host and the new one, it explains a lot. The old host had 1 CPU. The new one has 4 CPUs and faster ones at that. The old host had 2.5GB RAM and the new one has 6GB RAM. I do not believe the old host was fairly allocating server resources, because quite often the site was reaaaaaal slllooooooow, even though the number of visitors at that time was low. Other times, it was zippier despite the visitor count being high.
The new host probably has faster SSD drives and more recent motherboards - less bottlenecks to hold you back. Their pipe to the internet is likely to be a lot fatter too.
I am so pleased I made the move. It provides a nice stable platform for many years ahead. No more headaches!