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I am having a hard time understanding what differentiates the two types of slip. I understand that in a forward slip, the longitudinal axis is tilted with respect to the flight path, whereas in a side slip it is not. In a forward slip your track stays the same, whereas in a side slip it would (in the absence of wind) cause you to track toward the down wing.

My difficulty comes from the fact that the control inputs needed to enter both are the same: banking with opposite rudder.

Why then is the heading of the aircraft with respect to the flight path different, if the control inputs are the same? Is it just a question of degree, which is to say, the forward slip just requires more rudder?

There is no difference aerodynamically. The only difference is intention and presence of the wind. Airplane does not care about ground track, all it feels is movement through the air. Both side- and forwardslip makes airplane to fly slightly side-wise through the air, somewhere in the direction between nose and downwards pointing wing.

If there is no wind, your ground track will be exactly the same, between wing and nose, and you feel, that the nose points away from your track and call it forward-slip. If the wind is blowing and you manage right amount of the slip, you are mowing side-wise through the air, but the ground track (combination of wind and slip- movement) can be aligned exactly with the nose and you call it side-slip.

There is really no difference aerodynamically and thus no difference in applied controls neither.

The difference is in application. First, the forward slip (my favorite). The inputs (for a 172) are full rudder and opposite aileron as needed to hold straight path. This is cross controlled, but safe if you maintain a healthy margin above stall. I did mine at 65 knots. The purpose is to increase drag, producing a faster rate of descent for a given airspeed.

A side slip is used to hold your track in a cross wind. Here you aileron (drop wing) into wind as needed to stop being pushed sideways, using opposite rudder to control "weather vaning" of tail, and a tad more up elevator. This is where you want to be just prior to touchdown. Again cross controlled.

These are best practiced at altitude first, as different aircraft have different cross controlled stall characteristics. The 172 produced some tail oscillation in a forward slip with flaps down, but, especially at 65 knots, got me home every time.

Forward slip and side slip are the same. It's just the objective, degree of control input, and alignment of the nose is different. Forward slips are to lose altitude. Side slips are to compensate for lateral drift while landing.

In a forward slip your path through the air is more or less the same as before you started the slip. In a side slip your path is skewed laterally (kind of a forward slip with insufficient input).

Whether it's forward or side slip depends on the amount of control inputs. A forward slip has enough rudder to point the nose to the side, with opposite bank to prevent turning, so that your path continues straight ahead in the absence of a crosswind.

A side slip uses rudder to keep the nose aligned with the original heading, and with bank applied to let the airplane "slide downhill" so to speak, into wind, the result being the airplane tracks along the ground where the nose is pointed even though there is a crosswind.

Sideslip and forward slip are actually the same thing. They are both intentional cross controlled, unbalanced flight. The only difference is where the runway is oriented with respect to the aircraft's attitude, and the reason for executing the slip.

In what is being called a "forward slip", (used to lose altitude without gaining airspeed) the winds are approximately straight down the runway, so the aircraft attitude must be cocked to the side to keep the flight path lined up with centerline as the aircraft actually sideslips at an angle towards landing.

In a crosswind situation the reverse is true: The aircraft's flight path relative to the wind is into the direction the wind is coming from, (angle of bank into the wind to keep from drifting) but the aircraft is cross controlled with rudder to keep the nose lined up with centerline.

The aircraft's attitude with respect to the relative wind is the same in either case. Any time an aircraft is in unbalanced flight it is slipping both forward and sideways to some extent. The only difference is whether you are compensating for a crosswind or not, and where the runway is on your windscreen.

Forward slip, you're basically pulling the nose up and coming in pre-stall, but wings level, except for maybe some wind correction.

https://www.aopa.org/news-and-media/all-news/2018/july/flight-training-magazine/technique-forward-slip

In a side-slip, you're turning the plane much more sideways and using the side of the fuselage to create a draggy profile.

http://www.studyflight.com/side-slip/

I prefer side slip, especially if push the tail left, gives a great view out the pilot side of the plane. Not so good a view from the other side.

If I'm coming in to someplace new with no glide slope indicators, I tend to come in high, and make up for with sideslip to kill speed & lose altitude.