Can you explain the difference between SVC and LinearSVC in scikit-learn?
A regular SVM with default values uses a radial basis function as the SVM kernel.  This is basically a Gaussian kernel aka bell-curve.  Meaning that the no man's land between different classes is created with a Gaussian function.  The linear-SVM uses a linear kernel for the basis function, so you can think of this as a ^ shaped function.  It is much less tunable and is basically just a linear interpolation.

People are kind of hammering this question because you didn't provide a lot of information, but looking deeply into what you posted... this hits on some fundamental aspects of really understanding the details of bias and variance and the difference between linear and nonlinear basis functions in SVM.

Check out this image describing the four regions of high and low bias and high and low variance.  Obviously the best place to be is  low variance and low bias.



First lets assess variance -

Now take a look at your plots:


The nonlinear basis function has higher variance.  See how it is noisier than the linear kernel!  The linear kernel has lower variance.  See how it is less noisy!  

Now lets assess bias -

Which kernel is more accurate?  We can add the errors that you provided.  The nonlinear kernel has a total error of ~550+325=~875.  The linear kernel has an error of ~690+~50=~740.  So the linear kernel seems to do better overall, but they are pretty close overall.  This is were things get tricky!  

Putting it all together

See how the linear kernel did a poor job on 1's and a really great job on 0's.  This is pretty unbalanced.  Where as the nonlinear kernel is more balanced.  It kind of seems like the sweet spot might be to create a balanced model that doesn't have such high variance.  How do we control for high variance?  Bingo - regularization.  We can add regularization to the nonlinear model and we will probably see much better results.  This is the C parameter in scikit learn SVMs, which you will want to increase from the default.  We could also play with the gamma parameter.  Gamma controlls the width of the Gaussian.  Maybe try increasing that one slightly to get less noisy results e.g. a larger no-man's land between classes.

Hope this helps!