Does the slope of a regression between observed and predicted values always equal the $R^2$ of the original model?

As the title to my question says, I am confused as to when the $R^2$ of a model fit does not equal the slope of the regression between observed and predicted values.

I am trying to present model prediction statistics in a similar way to those presented in the summary figures of the Globcolor validation report (link) - (e.g. figure from page 53 of the .pdf): Here we see that they present the plot of observed versus predicted Chlorophyll concentrations, as well as statistics relating to its regression (e.g. the dashed line: $R^2$, $RMS$, $\alpha$ - intercept, and $\beta$ - slope).

My issue is that in my comparisons, I always get exactly the same value for the overall model fit $R^2$ and $\beta$-slope of the observed versus predicted regression.

Basic question: When (if ever) can these be different?

I have included a basic example of my problem in the following R script:

set.seed(1)
n <- 100
x <- runif(n)
e <- rnorm(n)
a <- 3
b <- 5
y <- a + x*b + e

#fit model
fit <- lm( y ~ x )

#plot regression
plot(x,y)
abline(fit)

#plot predicted versus observed
png("plot.png", units="in", width=5, height=5, res=400)
par(mar=c(5,5,1,1))
pred <- predict(fit)
plot(y, pred, xlim=range(c(y,pred)), ylim=range(c(y,pred)), xlab="observed", ylab="predicted")
abline(0,1, lwd=2, col=8)