I would expect soft-margin SVM to be better even when training dataset is linearly separable. The reason is that in a hard-margin SVM, a single outlier can determine the boundary, which makes the classifier overly sensitive to noise in the data.

In the diagram below, a single red outlier essentially determines the boundary, which is the hallmark of overfitting

To get a sense of what soft-margin SVM is doing, it's better to look at it in the dual formulation, where you can see that it has the same margin-maximizing objective (margin could be negative) as the hard-margin SVM, but with an additional constraint that each lagrange multiplier associated with support vector is bounded by C. Essentially this bounds the influence of any single point on the decision boundary, for derivation, see Proposition 6.12 in Cristianini/Shaw-Taylor's "An Introduction to Support Vector Machines and Other Kernel-based Learning Methods".

The result is that soft-margin SVM could choose decision boundary that has non-zero training error even if dataset is linearly separable, and is less likely to overfit.

Here's an example using libSVM on a synthetic problem. Circled points show support vectors. You can see that decreasing C causes classifier to sacrifice linear separability in order to gain stability, in a sense that influence of any single datapoint is now bounded by C.

Meaning of support vectors:

For hard margin SVM, support vectors are the points which are "on the margin". In the picture above, C=1000 is pretty close to hard-margin SVM, and you can see the circled points are the ones that will touch the margin (margin is almost 0 in that picture, so it's essentially the same as the separating hyperplane)

For soft-margin SVM, it's easer to explain them in terms of dual variables. Your support vector predictor in terms of dual variables is the following function.

Here, alphas and b are parameters that are found during training procedure, xi's, yi's are your training set and x is the new datapoint. Support vectors are datapoints from training set which are are included in the predictor, ie, the ones with non-zero alpha parameter.

Though very late, I don't agree with the answer that was provided for the following reasons:

• Hard margin classification works only if the data is linearly separable (and be aware that the default option for SVC() is that of a 'rbf' kernel and not of a linear kernel);
• The primal optimization problem for an hard margin classifier has this form:
• On the other hand, as you might see from the guide, the considered primal optimization problem in scikit-learn is the following:

This formulation - which in literature identifies the optimization problem for a soft margin classifier - makes it work also for non-linearly separable datasets and introduces:

• zeta_i, which measures how much instance i is allowed to violate the margin (the functional margin can be less than 1 in passing from the first to the second formulation);
• hyperparameter C, which is a portion of the "penalty" imposed to the objective function for permitting instances to have functional margin less than 1.

Eventually, as you might see in sklearn doc, hyperparameter C must be strictly positive, which enforces the idea that SVC() does provide soft margin classification. Here is another SO reference.