that's not how it works in cod
cod's engine is based on idtech and idtech has certain sweetspots which can be calculated by taking it's max framerate (1000) and dividing it with any integer. those sweetspots give certain advantages. the framerates which offer the best advantages are 125fps (1000/8), 250fps (1000/4) and 333fps (1000/3). these sweetspots offer higher firerates and further and faster jumps. so yes, in cod you have an advantage with a higher framerate. that is also why if you choose a 120fps cap in bo2 you get 125 fps. there is nothing more important in cod than having a steady framerate at any of those sweetspots. 63fps is also a sweetspot which is btw what you get when you limit the framerate at 60fps, but it barely offers any advantages.
if you get an unstable framerate or some bullshit like 90 fps you will feel a difference and it's not a pleasant one.
/edit
even though your GPU is horrendous the framerate in cod is more influenced by the cpu than the gpu because the engine is so effective that any recent gpu can handle it easily. it's the cpu bottlenecking it. well, your gpu might be at max load, that thing is a joke.