I think this problem goes deeper than I originally thought, so i created a Notebook that should be easy to read and make the problem visible (attached bug.nb)

Here it can be seen that with the 2nd assumptions where "d" is an Integer, the expression

f1[n] - f2[n] //FullSimplify

is recognized to be 0 (which is true). The expression

a - Sum[ Binomial[d, n] f1[n] , {n,0,d} ] //FullSimplify

is not recognized to be 0, but

a - Sum[ Binomial[d, n] f2[n] , {n,0,d} ] //FullSimplify

is recognized to be 0.

Edit

Definitions:

a:= (Exp[x] + Exp[-x])^d
f1[n_] := Exp[(2 n - d) x]
f2[n_] := Exp[-x]^(d - n) Exp[x]^n

Assumptions 1 ( //FullSimplify correctly yields all zeros ):

d >= 1 && d >= n >= 0 && Element[x, Reals] && Element[n, Integers]

Assumptions 2 ( //FullSimplify does not yield a - Sum[ ... f1[n] ...] == 0 ):

d >= 1 && d >= n >= 0 && Element[x, Reals] && 
 Element[{n, d}, Integers]

Attachments: