If then is tricky with ModelBuilder.
I think the most straightforward way to do this is to do the compare with a Python function inside a Calculate Value tool. Have your function return 1 or 0 and make the returned value a precondition to your append operation.
expression
test_same(r"%Featureclass1%", r"%Featureclass2%")
code block
def test_same(fc1, fc2):
nm1 = arcpy.Describe(fc1).name.lower()
nm2 = arcpy.Describe(fc2).name.lower()
if nm1 == nm2:
return 1
else:
return 0
