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The main difference between symbolic functions and symbolic variables is that symbolic functions are callable, i.e. you can evaluate the value of the function at any given argument via the parentheses operator:

sage: x(t) = sqrt(b^2 + c^2*t^2) - b
sage: x(0)
-b + sqrt(b^2)
sage: x(pi)
-b + sqrt(pi^2*c^2 + b^2)

whereas for a symbolic variable, you have to use the subs method:

sage: x = sqrt(b^2 + c^2*t^2) - b
sage: x.subs(t=0)
-b + sqrt(b^2)