What is the difference between a "model" and a "theory"? In my past questions I have used the terms "model" and "theory" interchangeably.  
So we have statements along the lines of  The Standard Model is our best theory of particle physics but I have also seen the X idea is a model, not a theory.
So why the distinction?  Is it as clear cut as the difference between a Theory and a Hypothesis? 
 A: From https://www.researchgate.net/post/What_is_difference_between_a_theory_and_a_model:

A theory is a set of statements that is developed through a process of
  continued abstractions. A theory is aimed at a generalized statement
  aimed at explaining a phenomenon.
A model, on the other hand, is a purposeful representation of reality.
As you can see, both share common elements in their definitions. What
  differs one from the other (in my opinion) is that one is aimed at
  generalized statements(theory) while the other is aimed as a helpful
  tool to understand specific phenomena(modeling).
Another way to link the two and point out differences is, a model is
  often used to describe an application of a theory for a particular
  case. Sometimes it involves a given set of initial and boundary
  conditions.
For example, the behavior of the Eiffel tower in an earthquake may be
  modeled by a finite elements computer simulation. The underlying theory employed could be the Prandtl-Meyers Stress-Strain relationship for elastic-plastic flow in metals and, of course,
  Newtonian mechanics.  In other cases, the term model is used more
  generally to mean some abstract representation or approximation to an
  underlying theory. In this sense, the P-M relationship above can be
  referred to as a "model" of the behavior of metals.

A: It is interesting to note, as a mathematician our language is (obviously) almost the $opposite$ of the answers given.  To use the vocabulary of model theory and meta-logic, a theory is a set of sentences which can be derived from a formal model using some rule of inference (usually just modus ponens).  So, for example, Number Theory is the set of sentences true about numbers.  But the model  is a structure together with an interpretation.
A: I was taught that the Standard Model was a misnomer; that it ought to be called the Standard Theory. I'm inclined to agree, though theories and models are both indispensable in science. 
Ultimately, the purpose of a model is provide local understanding of a particular phenomena. A model:


*

*Typically considers only fields, objects or quantities relevant to a particular phenomena

*Typically considers a particular energy scale.

*Provides local explanations of phenomena, often in terms of intuitive concepts or with metaphors (plum-pudding mode, billiard-ball model etc)

*"Truth" (i.e. scientific realism) is not the goal of modelling - understanding is the goal.


A theory, on the other hand, is supposed to be closer to the "truth":


*

*Typically broad in scope - considers many fields, objects and quantities relevant to multiple phenomena.

*Typically applies to many energy scales.

*Often lacks intuitive explanatory power - applying a theory to specific case may be complicated.

*"Truth" is an important goal - theories are supposed to be (approximations) to reality, rather than stories that help to understand a phenomena.


There is a reciprocal relationship between theories and models - scientists use both to develop their ideas. There are gaps in our understanding about the roles of models and theories, particularly in high-energy physics.
Indeed, returning to my opening remarks, it is unclear whether effective field theories, such as the Standard Model, play the role of theories or models, or something in between.
A: *

*the theory describes the phenomena i.e. the incident and the model describes the phenomena in a described way. 

*theory is a abstract and the model is simple 

*theory is literal and model is mathematical

*theory is based on certain assumptions whereas model doesn't assume anything 

*theory is unlimited whereas model is definite

