What is the difference between the forward transfer function and closed-loop transfer function?
I just wanted to make things a bit more clear here because it seems that the idea of open loop/closed loop/forward transfer function has got a bit mystified and does not seem exact even though it really is.
If you have a dynamical system with input \$u(s)\$, output \$y(s)\$ defined as:
$$ \frac{y(s)}{u(s)} = G(s)$$
Dynamical systems described with transfer functions are idealized, generalized and abstracted, many different systems can be described with the same transfer function. From the transfer function you can ideally find out everything you need to know about the system from the point of view of the control engineer, but that often is not a case.
Transfer functions can be stable and unstable:

Stable - all poles are negative

DC motor (shaft velocity, armature current)
Room temperature...


Unstable - at least one pole is positive or equal to zero

Inverted pendulum
Ball on plate
Segway, Onewheel,..



                                     
In the general, the transfer function's behavior, poles and zeros, time constants and characteristic frequencies are different then you want them to be and there for therefore you need a controller. There are two types of control you can apply to the physical system defined as the one above:

Open-loop control
Closed-loop control

Open-loop control
Open-loop control procedure does not rely on measurements of the controlled variables and assumes that system behavior is well known and deterministic, therefore it can be controlled without any knowledge what happens with the output value \$y(s)\$.
             
The complete open-loop transfer function(also known as forward transfer function) is no longer in between input \$u(s)\$ and output \$y(s)\$ but set point (reference) value of the output \$r(s)\$ and \$y(s)\$:
$$ \frac{y(s)}{r(s)} = C(s)G(s)$$
With the poles and zeros of the controller \$C(s)\$ you can tune the behavior of your complete system, even stabilize it in theory. In theory the perfect controller of the open loop procedure would be:
$$ C(s) = \frac{1}{G(s)} $$
But what happens in theory is that systems have uncertain stochastic disturbances \$d(s)\$, which you cannot anticipate. And more importantly you cannot compensate without measurement. These disturbances can be a simple as measurement noise, but can be much more complicated and harmful.
      
To be able to compensate the parts of the stochastic parts of the system you will need to introduce some kind of measurement. And therefore you need to &quot;close the control loop&quot;.
Closed-loop control
Closed-loop control is everywhere and it has well described and documented synthesis procedures and analysis frameworks. The following image shows simple general closed loop block diagram.

The complete transfer function of the closed loop is derived like this:
$$ d(s) = 0 $$
$$ y(s) = \Big[r(s) - y(s)\Big]C(s)G(s) $$
$$ y(s)\Big[1 + C(s)G(s)\Big] = r(s) C(s)G(s) $$
$$ \frac{y(s)}{r(s)} = \frac{C(s)G(s)}{1 + C(s)G(s)} $$
Usually, when you are designing the controller \$C(s)\$ you are setting the poles and zeros of the open loop transfer function, using Bode plot, Nyquist plot, root locus, compensation algorithms, loop shaping and similar.
The easiest way to understand this is if you look at the closed loop transfer function denominator.
$$ 1 + C(s)G(s) = 1 + G_{open\,loop}$$
What you usually do when you have a transfer function is that you evaluate roots of the denominator - the poles. If you want to know what the behavior of your new transfer function is going to be you have to solve the equation:
$$ 1 + C(s)G(s) = 0 $$
By placing the poles and zeros of the closed loop transfer function properly you will be able to get away with a lot of uncertain and stochastic influences in the system, such as:

Unknown disturbances
Unknown parameters
Unknown dynamics
System nonlinearity

You can try to follow some tutorials to understand better what the procedures are and what do you gain from using closed loop method.
Mathworks tutorials are great for these purposes.