Bootstrapping is any test or metric that relies on random sampling with replacement.It is a method that helps in many situations like validation of a predictive model performance, ensemble methods, estimation of bias and variance of the parameter of a model etc. It works by performing sampling with replacement from the original dataset, and at the same time assuming that the data points that have not been choses are the test dataset. We can repeat this procedure several times and compute the average score as estimation of our model performance. Also, Bootstrapping is related to the ensemble training methods, because we can build a model using each bootstrap datasets and “bag” these models in an ensemble using the majority voting (for classification) or computing the average (for numerical predictions) for all of these models as our final result.
Cross validation is a procedure for validating a model's performance, and it is done by splitting the training data into k parts. We assume that the k-1 parts is the training set and use the other part is our test set. We can repeat that k times differently holding out a different part of the data every time. Finally, we take the average of the k scores as our performance estimation. Cross validation can suffer from bias or variance. Increasing the number of splits, the variance will increase too and the bias will decrease. On the other hand, if we decrease the number of splits, the bias will increase and the variance will decrease.
In summary, Cross validation splits the available dataset to create multiple datasets, and Bootstrapping method uses the original dataset to create multiple datasets after resampling with replacement. Bootstrapping it is not as strong as Cross validation when it is used for model validation. Bootstrapping is more about building ensemble models or just estimating parameters.