It kind of annoys me how people don't actually understand the difference between the different super-saiyan forms other than "LUL HIS HAIR CHANGED HES STRONGER NOW" Let me explain:
Normal Saiyan: Their power is in a sense "dormant" because they can only raise their power level by "igniting" it. Their body is basically like a lump of coal, it contains alot of energy, but it can only release it a little at a time, increasing in intensity with how "hot" its burning. Think of Gokus Kaioken technique as the equivalent of pouring gasoline over a piece of coal, it burns hot and fast but only lasts a short time until returning to normal.
Super Saiyan: Their power has been unleashed. Continuing with the coal analogy, SSJ would be like if all the energy from the coal was released but kept inside of the body as pure energy, not a solid thing. It can now be released as intensely as needed, and is much more potent at that, as becoming a SSJ also unlocks hidden power in the body, like having a second piece of coal that you didn't know you had. The higher stages of SSJ (ASSJ and USSJ) are just the equivalent of a bigger piece of coal, having more energy at the expense of more volume (and slower movement).
Super Saiyan 2: Like the SSJ, their power is completely unleashed, but rather than just filling up their body with power that moves around randomly inside them, it is ordered and directed. The biggest change is in how they control their energy. Its like a bag of chips, when you first fill it up the chips take up a lot more space because they are all angled haphazardly around inside, but if they are all placed in vertically, you can fit in a lot more chips in less volume, they are easier to get out, and are less prone to breaking. Replace the word "chips" in that last sentence with "energy" and you have SSJ2. They can direct the alignment of their energy within them and organize it, to be used much more efficiently. The second biggest change is their speed, which, as a result of their new ability to direct their energy, increases exponentially. A noticeable visible change as a result of this form is their aura, which when SSJ2 can be seen flowing upwards, and their hair points straight upwards in line with the energy inside of them.
Super Saiyan 3: This is nothing more than an incredible increase in power. It is like when Trunks goes USSJ but without the bad bulking-up side effect. Instead of expanding their body, the power is condensed to allow for even more energy to fit inside their body and be used. Whatever excess energy can't be stored in the body is stored in the hair, which increases greatly in volume. The more worn-out a SSJ3 is, the smaller his hair becomes, which is evident when Goku goes SSJ3 for the second time on Kamis lookout, where his hair is smaller than when he did it for Buu. They can also expend their energy at a much higher rate, as their body (due to the increased power density) changes dramatically and is much sturdier, a better conduit for releasing energy. The drastically increased energy output is what makes it harder to maintain SSJ3 as long as the other states.
Super Saiyan 4: This state isn't so much an energy change as it is a mental and physical change. With the other 3 transformations, it was always a change in how the energy was used, but with SSJ4 its the vessel that carries it. Like with SSJ3 The body becomes much more durable, becoming almost indestructible, but one of the most notable changes is its Saiyan adaptability. Until reaching SSJ4, Saiyans have the ability to become stronger after recovering from battles. With SSJ4, their body adapts and changes during battle, repairing itself and making the fighter more or less immune to attacks that it has experienced before. It cannot negate directly damaging attacks such as a punch or energy blast, but things like freezing, and specialty attacks lose their effect after repeated use. It also increases the Saiyan's mental capacity, increasing reaction time, energy control, and their ability to assess and design a battle strategy and implement it. There is a remarkable energy increase with this transformation as well, resulting in a larger volume of hair. (Though differently styled than SSJ3)