There's a few options. One is to set up replication within MySQL, which will automatically copy data back and forth between the servers and keep them synchronized automatically, with a granularity of a few seconds as the data flies around. Downside is you have to expose at least the master server to the net to allow TCP connections.
Externally you can do regular mysqldumps on server A, copy to server B, load into mysql, and off you go. This will have a granularity of whatever time interval you run the dump/copy/load sequence in. Downside is that mysqldump locks tables as it's working. If you have a large database, you'll be locked out of serverA's tables while the dump progresses, and the locked out of serverB as the data's loaded. And loading a dump is much slower than doing the dump in the first place, as mysql does all kinds of internal housekeeping (key updates, table metadata updates, etc...) during the load.
For the third option, you could write a synchronizer that compares rows between the two databases and does updates as necessary. However, then you're slurping the contents of one server over to the other and doing row-by-row operations. Plus having to handle any dependent foreign key/child relation updates as well, and this will get complicated in a hurry. At that point you might be better off using the mysqldump option instead.
Of course, you should upgrade your MySQL to something more modern. v4.x is very outdated. You should be running at least a 5.0, preferably a 5.1 version instead.