Hi,

Having stayed at both, I think the lounge breakfast is better at the Hilton than the Conrad ( have not had the buffet breakfast). I liked the walk through the underground passage way from Shinjuku to the Hilton and the Shinjuku city park close by but also liked the overhead walkway and covered walkways from Shimbashi to the Conrad ( at the start of this month when I was at the Conrad I made detours via the Carretta shiodome whiere they had winter illuminations).
The difference in points between the Hilton and Conrad can be substantial though.

Regards

TBS