Is Odex really faster? Is an odex file different from the dex file generated and stored in dalvik-cache? 

Is it faster? More optimized? Or just the same?

Most people mention the benefit of improved boot speed after wiping the dalvik cache or on new boot, but no one ever discusses runtime performance of both files.
The DEX file stored in dalvik-cache is an optimized ("odex") file.  For historical reasons it carries the file extension ".dex".

The canonical documentation on dexopt is in dalvik/docs/dexopt.html.  It's a bit stale at this point but is a reasonable starting point.