Linear regression VS linear modeling Can I claim that linear regression and linear modeling are the same topics? If not, what is the difference?
 A: Comment made into an answer per suggestion of gung.
Linear modeling can have meanings, outside Statistics, well beyond the Wikipedia entry Linear Model in whuber's comment above. For instance, Linear Programming https://en.wikipedia.org/wiki/Linear_programming is the minimization or maximization of a linear function of several (could be millions) variables subject to linear constraints on those variables. Creation of the model to be solved by Linear Programming is considered to be linear modeling. 
Without Linear Programming (it is widely used in oil refining), the gasoline (petrol) you buy for your car would be more expensive, and transportation would cost more (aside from petrol cost). I would venture to say that Linear Programming (to include Mixed Integer Linear Programming) plays a far more important role in the U.S. and world economies than does linear regression, and is THE most important and greatest impact linear modeling which is performed. 
That said, I'm a nonlinear guy, so I see nonlinearity everywhere.  On the other hand, I sometimes see how to restrict linearity to cost functions (input data to optimization), and thereby still perform "linear modeling" and solution, even though  I have managed to get (sneak) significant and vital nonlinearity into the "linear" model.
A: From my point of view, linear regression is one kind of linear modeling. Thus, this modeling can refer to a full rank model (regression) or to a model not of full rank (experimental designs, for example). Modeling is a more general term, with several applications.
A: My impression is that the term linear regression (especially linear regression 'analysis') is used more often to explain relations while modeling is used more often in context of predictions and predictive models. 
