Many Oracle products can happily coexist on the same computer. Basically, you should install every Oracle product into its own directory (i.e. "Oracle Home"). I suppose you installed Client 10g over Client 9i; maybe you should clean up that mess (uninstall everything, delete directories, possibly registry entries) and start over again.
You might have certain issues - such as many TNSNAMES.ORA files you need to maintain, but that is easily resolved by setting the TNS_ADMIN environment variable.