I am trying to import data from one database in a server to a new table in a different server, using SQL Server import and Export Wizard. (SQL Server Management Studio 2012)
I bet that text columns aren't able to be inserted into varchar columns with the wizard. Depending on how big the table is, you could export the source to csv via the SSMS and then import it. That should work but if you have multiple tables to import you could add a linked server. Then you can just qualify the old table or the new table like this:
insert into [new_server].database.dbo.tablename
select * from old_table
I know SQL2000 is a pain for creating linked servers though which is what I'm guessing you are trying to export from since you have text columns.