What is the difference between toy models and normal models? Here is the short description of scientific model:

an imperfect or idealized representation of a physical system

And the definition of toy model:

a simplified set of objects and equations relating them so that they can nevertheless be used to understand a mechanism that is also useful in the full, non-simplified theory.

I don't understand the difference between them. Isn't any model used to "understand a mechanism that is also useful in the full, non-simplified theory"? The Ising Model is listed as a toy model, but isn't that natural when modelling a phenomenon: you start from a simple explanation, then improving it by adding more correction? And if it is just a toy model, why bother to expand it to higher dimensions? It even has application in neuroscience. I would say it's quite successful.
IMO even Standard Model can be seen as an toy model. Is that correct? And at what point it is not a toy model anymore? When it can give some quantitative results that agree with experiments? But then any model does have its own limitation, right?
 A: I don't think we should think about this in terms of definitions, and of a particular model being or not being a toy model, but rather it is a matter of the spirit with which a model is considered.
A model usually qualifies as a toy model when it is considered mainly not as a (however rough) description of reality, but as a simplified version of a more realistic model to study some of its features without all the complications. In doing this it is quite usual to go past clear unrealistic features.
The question of whether a particular model is a toy model in general is ill-posed. For example a model of spontaneous U(1) symmetry breaking in 2 dimensions can be considered as a toy model for spontaneous symmetry breaking in a more general context, or a rather realistic effective theory in condensed matter.

IMO even Standard Model can be seen as an toy model 

It could, but the point is that it is a quite complicated and realistic model (3+1 dimensions, all the right degrees of freedom etc) and it agrees incredibly well with experiments.
A: A toy model is simply a very simple model which nevertheless is able to explain qualitatively a certain phenomenon. 
A model should be able to explain natural phenomena in a quantitative way.
Also, a toy model can be fundamentally flawed, mathematically or physically, or totally unrealistic. A model instead should be mathematically consistent and not contradicting other established physical theories. 
The Ising model for example is a toy model. It is not realistic, since it does not take into account the full rotational symmetry of the ferromagnet. Also, it gives quantitative predictions which are not consistent with experiments (e.g., critical exponents do not agree with experiments).
The Standard Model is not a toy model. It is spectacularly consistent with experiments (only recently some experiments begin to show little discrepancies with theoretical predictions), it does not contain major mathematical inconsistencies, and is consistent with quantum mechanics, thermodynamics, and special relativity. 
A: I would say there are two essential distinctions.
1) A toy model is based on assumptions that we KNOW TO BE FALSE. And not just for the sake of simplification in the sense of "point masses" and "frictionless planes"... but assumptions that are more than idealizations for convenience, they are stripping the problem down to a cartoonified state that is not realistic in any meaningful way.
2) Most toy models are based on an analogy between the system of interest and some other well-understood system. 
