How Linear SVM Regression and Multiple Linear Regression different in terms of the regression result?

They starts from the same equation as below.

y = w*x + b

But they solve it differently. MLR specified the w and b by minimizing the square error whereas SVM specified w and b by minimizing the loss function defined by C and epsilon.

I am wondering if the result of regression is significantly different. I guess that if the given data set is clean and well-explained by input features, the resultant w and b between SVM and MLR will be close. Putting my original question differently, I don't find any reasons to use linear SVM regression over multiple linear regression.

Furthermore, if the SVR you're talking about is like the one described here (great resource there, by the way) then choosing a larger $$\epsilon$$ will yield a flatter regression line/hyperplane, whereas you don't have such a tuning parameter for regular ol' linear regression. This parameter enables you to try to fight overfitting by flattening the regression line.