In my oppinion, there are several different goals people have when lifting weights.
As you have mentioned in your question, there is look/size/strength. There are also others such as endurance/conditioning.
I think everything can really be put into two classes;
- People trying to attain 'the look'
- People trying to attain 'strength'
Note: there can and will be crossover of the two.
Each of these can be broken down even further. Before I get into that though, there is a little bit of a standard when we talk about rep ranges (keep in mind, this is my opinion from experience and research and it is also quite a simplified look at it):
1-5 Strength Training
6-8 Size Training (Hypertrophy)
8+ Endurance Training
People who train to look better are called body builders. You can't simply class all body builders together however, since they are also training towards different goals. Some things do remain the same in body building such as symmetry, low body fat/water during competition and higher muscle mass then the average person.
At one end of the spectrum you have the Mr Olympia's (Ronnie Coleman, Arnold Schwarzenegger), these are people who are looking to put on a ridiculous amount of muscle while keeping everything symmetrical and 'proportional'. They can walk around on stage at around 135kg at 3% body fat.
Then you have people who want to achieve an aesthetic look and train towards a physique category (Steve Cook, Lazar Angelo). There people want to put on a lot of muscle but keep it to a level where it's not overdone. Everything needs to keep to strict proportions such as a small waste with the V taper laterals.
Again, strength training can be broken down by what certain people have as their goal. The one thing that remains the same however is that they all obviously want to gain strength in some form.
At one end of the spectrum, you have the worlds strongest men and heavyweight power lifters. (Žydrūnas Savickas, Konstantīns Konstantinovs) These people can be 200kg monsters with ridiculously strong cores to assist the massive and heavy lifts they perform. They do not care about their body fat or proportions and only have the goal of getting as strong as possible.
Then you also have people who compete in other activities, one example could be MMA, where the person has to fall into a certain weight class but still wants to gain strength in order to have the edge over opponents of the same weight. (Georges St-Pierre, Jon Jones) There people keep their body fat to a minimum and while they still heavily train cardio, their lifting programs will mainly be aimed towards strength and conditioning rather than size.
In summary, there are several differences in how people train with different goals in mind. If there wasn't everyone would be training the same way. These things have been scientifically proven and tested by numerous athletes over the years so it is a known fact that certain training styles help certain goals.