Yes, but with proviso's
1] Same version of Thunderbird on each computer, eg if a very old version of TB is on the older computer, there may be significant changes to the newer TB on the new computer.
2] DO NOT delete anything on the old computer, or get rid of the the computer, until everything is working well with the new TB on the new computer.
3] I am unsure if email settings will transfer across. They should, but I have never done it, but if the old computer is set up with RR and TimeWarner then looking at the account settings on the old and inputting them into the new should be fine.
4] PATHS. The Path for Thunderbird on an older OS like XP is different to the PATH on a newer OS like TB. For example the Path on my Win 7 system for Thunderbird is;
C:\Users\MDFlax\AppData\Roaming\Thunderbird , with profiles being;
In XP, the Path was in Documents and Settings.
I hope that helps.