I used to apply K-fold cross-validation for robust evaluation of my machine learning models. But I'm aware of the existence of the bootstrapping method for this purpose as well. However, I cannot see the main difference between them in terms of performance estimation.

As far as I see, bootstrapping is also producing a certain number of random training+testing subsets (albeit in a different way) so what is the point, advantage for using this method over CV? The only thing I could figure out that in case of bootstrapping one could artificially produce virtually arbitrary number of such subsets while for CV the number of instances is a kind of limit for this. But this aspect seems to be a very little nuisance.

Both cross validation and bootstrapping are resampling methods. 


bootstrap resamples with replacement (and usually produces new "surrogate" data sets with the same number of cases as the original data set). Due to the drawing with replacement, a bootstrapped data set may contain multiple instances of the same original cases, and may completely omit other original cases.
cross validation resamples without replacement and thus produces surrogate data sets that are smaller than the original. These data sets are produced in a systematic way so that after a pre-specified number $k$ of surrogate data sets, each of the $n$ original cases has been left out exactly once. This is called k-fold cross validation or leave-x-out cross validation with $x = \frac{n}{k}$, e.g. leave-one-out cross validation omits 1 case for each surrogate set, i.e. $k = n$.
As the name cross validation suggests, its primary purpose is measuring (generalization) performance of a model. On contrast, bootstrapping is primarily used to establish empirical distribution functions for a widespread range of statistics (widespread as in ranging from, say, the variation of the mean to the variation of models in bagged ensemble models).
The leave-one-out analogue of the bootstrap procedure is called jackknifing (and is actually older than bootstrapping).
The bootstrap analogue to cross validation estimates of generalization error is called out-of-bootstrap estimate (because the test cases are those that were left out of the bootstrap resampled training set). 





  [cross validation vs. out-of-bootstrap validation] However, I cannot see the main difference between them in terms of performance estimation.


That intuition is correct: in practice there's often not much of a difference between iterated $k$-fold cross validation and out-of-bootstrap. With a similar total number of evaluated surrogate models, total error [of the model prediction error measurement] has been found to be similar, although oob typically has more bias and less variance than the corresponding CV estimates.

There are a number of attempts to reduce oob bias (.632-bootstrap, .632+-bootstrap) but whether they will actually improve the situation depends on the situation at hand.  

Literature:


Kohavi, R.: A Study of Cross-Validation and Bootstrap for Accuracy Estimation and Model Selection, Mellish, C. S. (ed.) Artificial Intelligence Proceedings 14$^th$ International Joint Conference, 20 -- 25. August 1995, Montréal, Québec, Canada, Morgan Kaufmann, USA, , 1137 - 1145 (1995).
 Kim, J.-H. Estimating classification error rate: Repeated cross-validation, repeated hold-out and bootstrap , Computational Statistics & Data Analysis , 53, 3735 - 3745 (2009). DOI: 10.1016/j.csda.2009.04.009
Beleites, C.; Baumgartner, R.; Bowman, C.; Somorjai, R.; Steiner, G.; Salzer, R. & Sowa, M. G. Variance reduction in estimating classification error using sparse datasets, Chemom Intell Lab Syst, 79, 91 - 100 (2005). 





  The only thing I could figure out that in case of bootstrapping one could artificially produce virtually arbitrary number of such subsets while for CV the number of instances is a kind of limit for this.


Yes, there are fewer combinations possible for CV than for bootstrapping. But the limit for CV is probably higher than you are aware of. 
For a data set with  $n$ cases and $k$-fold cross validation, you have 


CV  $\binom{n}{k}$ combinations without replacement (for k < n that are far more than the $k$ possibilities that are usually evaluated) vs. 
bootstrap/oob $\binom{2 n - 1}{n}$ combinations with replacement (which are again far more than the, say, 100 or 1000 surrogate models that are typically evaluated)

