Why could query performance improve enormous after db restore on new server with worse hardware?
The amount of memory available to the SQL Server is an important input to the choice of execution plan made by the query optimizer.
It is certainly possible that the new SQL Server, with its reduced memory, is encouraging plans without memory-consuming operations (e.g. sorts and hashes) that happen to be 'better' for the type of queries you are executing.
If this is indeed a primary factor, reducing the amount of memory available on the original server would also likely increase its performance. Typically, this would be unusual, assuming statistics are good and the queries usually get reasonable cardinality estimates. Nevertheless, it might explain what you are seeing; a detailed analysis would be required to be sure.
To answer your final question: yes statistics are a persistent part of the database and are included in a backup/restore.
Update
Based on the new information in the question concerning the vastly improved I/O performance of the second system, yes, that is most likely the reason.