The reason "show tables;" works is because mysqld will scan the database directory for .frm files only. As long as they exist, it sees a table definition.
If you imported the data into MySQL and this error message happens, the first thing I would immediately do is run this command: (BTW This is MySQL 5.1.45, but works in MySQL 5.x anyway)
mysql> show engines;
+------------+---------+----------------------------------------------------------------+--------------+------+------------+
| Engine | Support | Comment | Transactions | XA | Savepoints |
+------------+---------+----------------------------------------------------------------+--------------+------+------------+
| InnoDB | YES | Supports transactions, row-level locking, and foreign keys | YES | YES | YES |
| MRG_MYISAM | YES | Collection of identical MyISAM tables | NO | NO | NO |
| BLACKHOLE | YES | /dev/null storage engine (anything you write to it disappears) | NO | NO | NO |
| CSV | YES | CSV storage engine | NO | NO | NO |
| MEMORY | YES | Hash based, stored in memory, useful for temporary tables | NO | NO | NO |
| FEDERATED | NO | Federated MySQL storage engine | NULL | NULL | NULL |
| ARCHIVE | YES | Archive storage engine | NO | NO | NO |
| MyISAM | DEFAULT | Default engine as of MySQL 3.23 with great performance | NO | NO | NO |
+------------+---------+----------------------------------------------------------------+--------------+------+------------+
8 rows in set (0.00 sec)
If the server you imported the data into says InnoDB is disabled, then you have a big problem. Here is what you should do:
1) Drop all the Data from the New Import DB Server
2) Cleanup InnoDB Setup
3) run SHOW ENGINES; and make sure InnoDB is fully operational !!!
4) Reload the mysqldump into the new import server
Give it a Try !!!