Prof. Hajme Nakamura had asked the same question to HH MahA SwAmy of Kanchi when had come to have His dharshan - "When the bhAshyams of AchAryAl talk
about advaitham, how is that His successor (i.e. PeriyavA) is wearing vibbhuthi (basmam) and doing Chandra moulIswara pUja in the math daily ?"
SrI PeriyavAl's lucid and deep answer to this and also on the smArtha sampradhAyam and as to why it is generally asscociated with saivaite tradition, albeit wrongly, is worth reading:
"Sankara sampra dhAyam", Deivathin Kural (in Tamil), Vol II, pp: 119-157, Second Edition (1980).
[Note: The page numbers may slightly vary in the subsequent six editions of this volume]
When I started reading the chapter on Sankara SampradhAyam , to make a summary, I found that a lot of details of His discourse would be lost
in this exercise. It appeared that the best way to present His thoughts would be to translate the entire chapter. Ofcourse, the loss due to my translation is unavoidable.
I'll attempt to post a couple of pages of translation every alternate day. At this rate, I hope, this would go for a couple of weeks.
The copy right of this original book is with the publisher - Mr. Thirunavukkarasu. Hope, he wouldn't mind this free-lance translation, purely for the sake of members of advaita list.
In the running text, wherever my notes are inserted they are enclosed by square braces.
 HH Sri Chandhra sEkharEndhra Saraswathi Swamiji, 'Sankara SampradhAyam'. Deivathin Kural (in Tamil), Vol. II, Second Edition, ed. rA. Ganapathi, Vanathi padhippagam, Chennai, pp: 119-157 (1980).