Sufficiency of the sample Given a sample $X_1,.....,X_n$ with common pdf $f(x;\theta)$, it is clear that the sample $\boldsymbol X$ itself is a sufficient statistic for $\theta$. 
I am aware of the meaning of "containing all the information about $\theta$" that is the main feature of a sufficient statistic.
My question is: why is it so crucial to find a sufficient statistic that is "shorter" than the sample? What would be the problems related to the use of the sample as a sufficient statistic in statistical inference? 
I hope I am being more clear.
 A: In completion to the previous answer, a word of warning: it is only within a very limited class of probability distributions that there exist sufficient statistics that allow for a massive reduction in dimension. 
This constraint is known as the Pitman-Koopman-Darmois lemma and is expressed as follows on Wikipedia:

According to the Pitman–Koopman–Darmois theorem, among families of
  probability distributions whose domain does not vary with the
  parameter $\theta$ being estimated, only in exponential families is
  there a sufficient statistic whose dimension remains bounded as sample
  size increases. Less tersely, suppose the $X_i,\ i = 1, \dots, n$ are
  independent identically distributed random variables whose
  distribution is known to be in some family of probability
  distributions with parameter $\theta$. Only if that family is an
  exponential family is there a (possibly vector-valued) sufficient
  statistic $T(X_1, \dots, X_n)$ whose number of scalar components does
  not increase as the sample size $n$ increases.

This result implies that outside exponential families (like the Normal, Poisson, Binomial distributions), there is no possibility of a reduction in dimension from the $n$ original observations to a sufficient statistics. Most often, the "only" reductive sufficient statistic is the order statistic $(X_{(1)},\ldots,X_{(n)})$, where the observations are ordered. This is of the same dimension as the data, the only noise removed being the vector of the ranks.
A: There are a couple of answers to your question.
First, consider the idea of "data reduction."  In general, if we can summarize our data using fewer numbers, this may be preferable to using the complete data.  If we want to estimate the mean of a population, we can report our sample mean and we lose no information.  (One could argue that the sample variance should also be reported - we could report both and we're still only reporting two numbers rather than our entire data set.)  This allows for ease of reporting.  If you're writing a paper for publication or presenting to a company, it isn't feasible to show all of your data, but the sample mean will be able to convey a the same amount of information regarding the population mean as all of your data would.
In statistical inference, minimal sufficient statistics (sufficient statistics that cannot be reduced any further) have compelling properties that may make further inference easier.  For example, minimal sufficient statistics have relationships with complete statistics and conclusions can be drawn from these relationships more easily than directly proving, for example, that a statistic is complete or minimal sufficient.  I would refer to a more theoretical resource (i.e. Casella & Berger's Statistical Inference or online resources) that more explicitly detail the benefits of minimal sufficiency.
