Database Synchronisation Manually
There are a couple of ways I could think I would approach this issue:


If the changes made by me in the old database are reproducible easily to the new database, I'd go for it first. And in most of the cases if you follow Drupal best practices like using update hooks, alter functions, etc. they could be easily done to the new database by (sometimes) running only update.php
So, I'd recommend to move all the custom changes to the update hooks if you have time. Because, considering that the site is live, you could expect more changes to the database. And your code should be flexible enough; unless off course if you could convince the client to stop updating the live site.
If you are sure no database level changes would be done (which I doubt, Drupal being a database driven framework), you could use some tools for comparing the changes between the two databases. Check this stackoverflow question for list of tools that could be used. Take a diff of the database, find out the tables which have differences. From the tables, find out which entities are added/removed from the database. And then make the changes to as required.
I would not recommend this method over the first.