The 5th gen iPod "classic" was the first "video" iPod. It's video capability is more limited, and supported video resolution is lower, typically only 320x240. Even if the video is standard def (SD), and not HD, it is probably at least 640x480.
Also, current iTunes Store video content (such as TV shows and movies) uses DRM (copy protection). The 5th gen iPod was designed before the iTunes Store started selling video content (as it does now); I don't think it supports the DRM technology.
If you have video content that is not DRM'd and plays in iTunes (such as video podcasts), you may be able to use the convert command in iTunes. Select the item on a list in iTunes. From the iTunes menu bar
File -> Create New Version -> Create iPod or iPhone Version
That version may play on your iPod, so iTunes will sync it.