You can only use iTunes to sync to an Apple device e.g. an iPhone, you cannot use Windows Media Player to do this.
You can however 'share' the same music files between iTunes and Windows Media Player. You would then be able to use iTunes to do the syncing, and Windows Media Player for locally playing tracks including in Windows Media Center. To do this you need to use a format both iTunes and Windows Media Player can use. For Windows Vista, 7, 8, and 8.1 this could be either MP3 or AAC, or if you add a couple of extra (free) components it could also be Apple Lossless. For various reasons AIFF, WAV, WMA, and FLAC are not suitable for sharing. See my article here about this http://jelockwood.blogspot.co.uk/2013/06/using-apple-lossless-aka-alac-in-window s.html
Note: As of Windows 10, Windows Media Player now supports Apple Lossless as standard and does not need the two additional components to achieve this.
Once you have Windows Media Player able to play MP3, AAC, or Apple Lossless then the best approach is to add all the music via iTunes, do the syncing via iTunes, and set Windows Media Player monitor the iTunes folder but not to copy or move files. Windows Media Player should then add entries for any tracks to the WMP library while leaving things under the control of iTunes. This approach is best because it is not possible to do the reverse i.e. have iTunes monitor the WMP library folder.
Note: Windows Media Player will even see and use the embedded album artwork in AAC and Apple Lossless files as well as of course standard MP3 files.