Streaming Algorithms: Motivations for estimating frequency moments
$F_0$ counting (or estimating distinct elements, or "cardinality estimation") is very useful. 

Example: when you're doing profiling at the router level, you often want to estimate functions of distinct IP addresses, and since you can't just maintain counters for each possible address, $F_0$ counting turns out to be quite useful.

$F_1$ counting, or frequency estimation is also very useful:

Example: There's work on building statistical language models that involve counting frequencies of pairs of words, triples of words, and so on. You can't maintain counts for all such triples, so you often want "frequent" ones, or "heavy hitters".