I have never even thought of saving my SMS messages from one device to another; I have always assumed you will need to start over once you transfer devices.
There once was an option in older phones to save SMS messages to the SIM card- but it was a lot of work and I haven't seen that option in a few years. If you are using a SIM in both phones, this may be an option.
My Phone Explorer appears to be software specifically for Sony Ericsson- but you may want to contact them to see if they have plans on making an Android version.
There are backup/restore apps like SMS Backup or Sprite Backup, but these are made just to backup the device you are using and not for transferring between different devices.
Worst case, if you have a software development/database friend, maybe they can figure out how to convert the MyPhoneExplorer database into another format that can be imported into MySMSBackup or a similar application on Android.
Update: I just saw in the market there is a 'Sprite Migrate' beta application; If you can find a SE version of this to export your data this may be what you are looking for...