We present a statistical analysis of the X-ray bursts observed from the 2002 June 18 outburst of the anomalous X-ray pulsar (AXP) 1E 2259+586, observed with the Proportional Counter Array (PCA) aboard the Rossi X-Ray Timing Explorer. We show that the properties of these bursts are similar to those of soft gamma repeaters (SGRs). We find the following similarities: the burst durations follow a lognormal distribution that peaks at 99 ms, the differential burst fluence distribution is well described by a power law of index À1.7, the burst fluences are positively correlated with the burst durations, the distribution of waiting times is well described by a log normal distribution of mean 47 s, and the bursts are generally asymmetric, with shorter rise than fall times. However, we find several quantitative differences between the AXP and SGR bursts. Specifically, the AXP bursts we observed exhibit a wider range of durations, the correlation between burst fluence and duration is flatter than for SGRs, the observed AXP bursts are on average less energetic than observed SGR bursts, and the more energetic AXP bursts have the hardest spectra-the opposite of what is seen for SGRs. Unlike the case of SGRs, we find a correlation of burst phase with pulsed intensity. We conclude that the bursts are sufficiently similar that AXPs and SGRs can be considered united as a source class, yet there are some interesting differences that may help determine what physically differentiates the two closely related manifestations of neutron stars.