Are there tools to perform data comparison between two different schemas?
It's not impossible to compare two different schemas, it's a calculation of how confident you are in the result.  I've essentially borrowed from Bank Reconciliation techniques

Important: This reconciliation isn't about making sure the destination exactly matches the source in it's data context (there's a reason you're migrating to a new system), but you do need to explain any discrepancies!

Basis:


Identify metrics that you will use to measure (ie: total number of users, sum of all their ages, list of user id's and their postcodes...) I try to use several metrics from: Totals, Averages &amp; sample/detailed records
Dump this data into a common location (via views/reports/whatever is reasonable)
Compare your data and make sure any discrepancy can be explained


I've generally split the comparison into several methods (in particular for the detailed metrics):  


By Table: Usually using the Source Table as the definition, creating a set of views on the destination DB to try and reproduce the Source Table Data
By Object: Depending on the DB and what you're storing it might make more sense to create a holistic view of the objects (ie: the user) across several tables, similar to above creating a set of views to return that object as a set of results
By Reports: If the Source DB has a good comprehensive set of Management Reports, another technique is to reproduce those reports with the exact same formatting


Techniques:

No matter what method I've used to produce what will be compared, I end up with a set of files/views/dbs that hold the reconcilable Source and Destination data, then depending on the medium I can use one of the commonly available tools to compare them.  My own preferences are:


File Comparison
Create two different folders for SourceDB and DestinationDB (time/version stamped so I know when I did the reconciliation) and dump out the output of my metrics as relevantly named files, then use a comparison tool (such as CSDiff) to identify differences
Excel Comparison
In particular when dealing with Management reports, I'll create a set of Excel of workbooks to compare the reports (in essence making use of VLookups and comparing totals)
Data Comparison
Rather than outputting the reconciliation data to files or reports, output them to separate DBs, then use something like Redgate SQL Data Compare to compare the DBs


Other tools:

Not tried any of these, but a quick google search on 'Data reconciliation tools' gives:  


n-Tierfs Data Reconciliation Software
Testing Performance Data Reconciliation


There's others, but those were the ones that looked reasonable