Why do different pain killers have different effects on people?
Short answer: different people have different amount of active receptors. 
In treatment, combination scores of Pharmacodynamics and Pharmacokinetics determine the final effect of the drug. 
Receptors determine many effects of the drug in many pathways. 
Different people also sense pains differently (Psychology). 

Review answer

The purpose of treatment is to relieve pain and maintain function. 
Your question is biased. 
You cannot only concentrate on painkilling in maintaining health. 
These both are the reasons for the complains of the patient, not only the pain.

For instance, in rheumatoid arthritis, response to therapy can be quantitated using many measures including American College of Rheumatology system values ACR20, ACR50 and ACR70, which denotes the percentage of patients showing an improvement of 20%, 50% or 70% in a global assessment of signs (maintaining function) and symptoms (pains).

Each patient has own health, different from one another. 
Our body adapts to the environment and individual conditions of the body to maintain homeostasis. 
Receptors adapt for instance. 
They can be active or inactive - in short-run and long-run - again depending on the conditions at hand. 

Painkillers i.e. analgesics have different properties:


anti-inflammatory effect - acute and chronic conditions (inflammation is the major mechanism under many pathologies) e.g. nonsteiroidal anti-inflammatory drugs (NSAIDs, please, see this answer about the particular mechanisms and how different people have different effects from NSAID painkillers) and glucocorticoids (most) 
symptoms relieving specific drugs e.g. disease-modifying antirheumatic drugs (DMARDs)
anti-platelet effect - e.g. the older you get, the more platelets stick together.
anti-pyretic effect
...


which all can be toxic. 
Note that many drugs alone or/and as combination can work as painkillers i.e. pain relievers. 
Aspirin for instance has both anti-inflammatory and anti-platelet properties. 
However, it is rarely anymore used as anti-inflammatory. 
The anti-platelet property is dependent on the exact dosage of the administration. 
Aspirin's mechanism of action is to inhibit platelet COX which antiplatelet effect lasts 8-10 days (life of the platelet). 
In other tissues, synthesis of new COX replaces the inactivated enzyme so that ordinary doses have a duration of 6-12 hours. 
Please, review any Pharmacology -textbook for more info about aspirin.

Each these drug has own Pharmacodynamics and Pharmacokinetics


Pharmacodynamics answers to the question What drug does to the body? It stimulates some receptors, activates some pathways, ...
Pharmacokinetics - What does body do to the drug? It metabolises it (enzymes, receptors). It distributes it. It excretes it (kidneys, feaces).  


In treatment, you consider what is the target organ. 
You need to think what is causing the dysfunction and the pain. 
You try to restore the function and relieve pain. 
The component of drug needs to reach the target tissue e.g. your pancreas' beta cells do no produce insulin so your blood glucose is high. 
Complications of this are polyuria and eventually exodus if untreated. 
Insulin is injected into the fatty tissue. 
We do not have long-term acting insulin administered orally - our metabolism start to break the drug so it does not have wanted treatment. 
If insulin administered to the muscle, the time of action is too times less, again because muscle is metabolising the insulin i.e. a chain of peptides (protein).

No all symptoms and diseases have painkillers. 
For instance, prehemorrhoid and some types of itching related to  posthemorrhoids. 
However, for both, there are some special salvas for proplylaxis but they are not complete.

Now, you can start to read something in SuperBest's answer about host's physiological variables and addiction/tolerance which alters the mechanisms (receptors) of pathways in Pharmacodynamics and Pharmacokinetics.

Sources


Basic and Clinical Pharmacology, 11th edition, 2009, Bertram Katzung.
My notes in Pharmacology classes during 2014.