Why aren't the twin organs like both the eyes, ears, hands and legs lying both sides of a human body identical?

Body development is a complex process in which the interaction of a huge amount of molecules and the environment give rise to the final product.

In such a complex system is not surprising that some chaotic dynamics exists, and if so, small perturbations during the process give rise to not so small difference in the final product.

What is surprising is that, even with such a big complexity, the final product (like the two eyes) is a stable state. So I think the main question is not what make them different, but how the development process achieves to generate symmetry.

The topic you're looking into is fluctuating asymmetry, a broad field of study. Among the free reviews this one is a bit old but well on topic. The general idea is that there is "developmental noise", which is opposed by mechanisms for developmental stability. The review covers a variety of conclusions people try to read in to these studies, often about the stresses on the organism or the mechanisms by which development is "canalized" to reach a specific goal.

The nature of the noise is usually presumed rather than proven, since many different factors are presumably adding up to make small differences. We may assume (though I'm not always sure it is true) that mRNA, protein, and signalling molecules diffuse to reach their targets, which is to say they are subject to Brownian motion with random variations in timing. The need for multiple factors to get together for a complex process like transcription or translation means that the number of copies of a protein produced under a given circumstance will vary. An example of this in hypertrophic cardiomyopathy has been proposed to contribute to disease progression, due to random bursts of transcription in individual heart cells affected by the disease.

The other answers talked about how asymmetry happens. Here I say my idea on why asymmetry could be beneficial. Focusing on left and right brain lobes, you see that the difference between these lobes give new functionalities to brain and these new abilities could be beneficial during evolution of the taxa. I think asymmetry which looks like stochastic, maybe in thousands and millions of years, little by little (and some times suddenly) make new (useful and harmful) inheritable functions.