As @Peter Ellis has pointed out, getting differences between dates is a matter of some difference function or an easy subtraction, depending on your software.
For a small number of dates, clusters should be evident in any univariate distribution display.
For a larger number of dates, plotting (cumulative) number of events so far can identify cluster centres. If the pattern is like random dates from a uniform distribution, expect a line of constant slope with trivial fluctuations. If there are clusters, expect a staircase or stepped effect. (This graph can also show shifts in average frequency.)
Periodicities are alluded to here. Plotting occurrences against time of day, week, month, year or whatever is germane may help too.
All that said, the tougher questions have an answer here:
How can I group numerical data into naturally forming "brackets"? (e.g. income)
but I haven't searched for an R implementation. The code for the Stata implementation mentioned there is in the public domain and mostly written in Mata, a C-like language within Stata with strong support for matrices.
Note I've seen people confused by this, so I will add what is evident on a little thought. With dates, treated numerically as time lapsed since some origin, mean, minimum, maximum, median, etc. also have an interpretation as dates or as positions on a date line. Standard deviation, IQR, range are durations or elapsed times, not dates. With these simple caveats, ordinary functions, commands or routines for descriptive statistics are useful for dates, treated (if necessary, converted or coerced) as numeric variables. Even if you are smart on this, your software may be a little dumb.