Quantifying diversity of bird species I’m wondering if anyone could recommend a diversity index to look at the differences in breeding bird territories between two sites. These are my data - the numbers represent the number of breeding pairs of each species:
c(4L, 3L, 14L, 1L, 4L, 11L, 2L, 1L, 2L, 7L, 1L, 5L, 7L, 3L, 1L, 2L, 1L, 1L, 4L, 1L, 1L, 4L, 2L, 4L, 3L, 3L, 1L, 1L, 1L, 14L, 3L, 14L) -> site1

c(27L, 25L, 21L, 23L, 1L, 2L, 26L, 3L, 1L, 16L, 3L, 12L, 29L, 2L, 5L, 1L, 17L, 2L, 6L, 19L, 6L, 3L, 3L, 5L, 1L, 1L, 8L, 7L, 1L, 2L, 5L, 2L, 6L, 2L, 8L, 1L, 5L, 1L, 25L, 1L, 1L, 4L, 1L, 26L, 7L, 1L, 1L, 13L, 1L, 30L, 34L, 13L, 2L) -> site2

I’ve tried Simpson’s index in R:
library(vegan)

diversity(site1, index = "simpson")
diversity(site2, index = "simpson")

But the diversity index is very similar for both sites (0.94 and 0.96 respectively) and I would say site2 is considerably more diverse than site1.
Are there any functions that are more sensitive when quantifying diversity?

Just about any general book on ecological methods has a section on diversity measures and there are indeed several dedicated monographs on diversity in ecology alone, to say nothing about related literatures. Ecologists can get quite agitated and even angry about which measures are good and especially about if any is best. Much of the literature is naive mathematically or scientifically, although as an outsider I won't give precise references, beyond hinting that some popular accounts are shockingly muddled.
You are too pessimistic about Simpson's measure which shows in your case for both sites a high probability of random pairs of individuals being different species.  However, small differences can be important. I suggest that the Simpson measure is usually better thought of on a different scale.
You are evidently using Simpson's measure as the complement of $R := \sum_{i=1}^S p_i^2$, namely $1 - R$ or $1 - \sum_{i=1}^S p_i^2$. Here the notation is that there are $S$ species and the typical species occurs with probability $p_i$. The $R$ notation is suggested by the term repeat rate, one of many terms in the jungle.
A reference case is $S$ species occurring equally frequently, for which  $R =
\sum_{i=1}^S p_i^2 = \sum_{i=1}^S (1/S)^2 = S/S^2 = 1/S$. This motivates $1/R$ as a numbers equivalent, namely an equivalent number of equally common species. I get that measure to be about  16.1 and   24.0 for your two sites, which is a fair contrast.
Similarly, consider Shannon entropy $H := \sum_{i=1}^S p_i \ln(1/p_i)$ which for $S$ equally common species is $S (1/S) \ln S = \ln S$. Hence $\exp(H)$ is a numbers equivalent. For your sites that gives 21.6 and  30.8.
Simplest of all is just the number of species present, $S$ itself, which for your sites are 32 and 53.
In general, $(1/R) \le \exp(H) \le S$. The measure of those three that responds best to the occurrence of rare or unusual species is number of species!
These measures are all part of a family $\sum p_i^a (\ln 1/p_i)^b$. For  Simpson's measure as a sum of squared probabilities or repeat rate $a = 2, b = 0$; for entropy $a = 1, b=1$; and for the number of species $a=0, b=0$. This easy unification was published by I.J. Good in 1953. There are fancier families associated with the names of C.R. Rao and M.O. Hill, but that Good family is simpler.
There are many other measures in the literature.
If you prefer $\log_2$ or $\log_{10}$ for calculating entropy, the corresponding numbers equivalent is then $2^H$ or $10^H$.
Note: This answer treats the data literally, as being say results from comparable sampling effort. The problem that the more you observe, the more you will find, and other more biological points, are well covered in other answers.
All that said, any idea that a single measure can capture all the information here is likely to prove disappointing. But that is true of any summary statistic.
