  We show that when KP (Kadomtsev-Petviashvili) $\tau$ functions allow special
symmetries, the discrete BKP equation can be expressed as a linear combination
of the discrete AKP equation and its reflected symmetric forms. Thus the
discrete AKP and BKP equations can share the same $\tau$ functions with these
symmetries. Such a connection is extended to 4 dimensional (i.e. higher order)
discrete AKP and BKP equations in the corresponding discrete hierarchies.
Various explicit forms of such $\tau$ functions, including Hirota's form,
Gramian, Casoratian and polynomial, are given. Symmetric $\tau$ functions of
Cauchy matrix form that are composed of Weierstrass $\sigma$ functions are
investigated. As a result we obtain a discrete BKP equation with elliptic
coefficients.
