Interpretation of the hazard rate and the probability density function The books of survival analysis state that
$f(t) = \lim_{\Delta t\rightarrow 0}\frac{P(t < T \leq t + \Delta t)}{\Delta t} $
$h(t) = \lim_{\Delta t\rightarrow 0}\frac{P(t < T \leq t + \Delta t|T \geq t)}{\Delta t} = \frac{f(t)}{S(t)}$
I know $h(t)$ is a conditional probability and commonly used to describe the failure.
But $f(t)$ is also a metric of failure.
Can we use it to describe the failure?
If yes, what's the difference between them?
 A: Think of f(t) (or really, the expectation of f(t)) as life expectancy at birth - the only condition is that the subject is alive at time T = 0. f(t) can be very useful, it is the distribution of failure times.
h(t), however, is often more interesting. As someone who has lived quite a few years past birth already, f(t) seems less useful whereas h(t) has immediate relevance.
From a modeling perspective, h(t) lends itself nicely to comparisons between different groups. In a hazard models, we can model the hazard rate of one group as some multiplier times the hazard rate of another group. With Cox Proportional Hazards we can even skip the estimation of the h(t) altogether and just estimate the ratios. 
Of course, this will result in different f(t) distributions for the groups, but whereas this multiplier is a convenient way to summarize the differences between groups both in a model and in communication, I wouldn't know how to summarize the resulting f(t) distributions of the two groups other than to plot them. While one group's hazard rate can always be twice that of another group, no such simple construction will work for f(t) since f(t) must always integrate to 1.
