Filters vs nets in topology Nets are a natural generalization of sequences in arbitrary topological spaces. Using the language of nets we can extend intuitive, classical sequential notions (compactness, convergence, etc.) to arbitrary spaces. Example of using: Reed, Simon "Methods of Modern Mathematical Physics: Functional Analysis".
There is an alternative (but essentially equivalent) language of filters. For example, Bourbaki use it a lot in his "General Topology".
IMHO, filters are completely unintuitive compared to nets, but many authors besides Bourbaki still uses filters to explain things. So, what are pros and cons of filters versus nets.

Nets involve a partial order relation on the indexing set, and only a part of the information contained in that relation is relevant for topological purposes.  The relevant part is just what is retained when one passes from the net to the associated filter.  So, in a sense, the use of filters discards irrelevant information that is present in nets.  
I believe I learned about nets before filters, so my preference for filters is probably not based on timing. It's more likely to have resulted from a (congenital?) preference for simplicity and for discarding or at least ignoring irrelevant information.  I agree, though, that after one learns basic notions in the context of sequences, nets, being rather similar to sequences, will be more intuitive, until one encounters subnets.
