In general, I see two reasons. First, it looks nicer. Second, it helps see patterns. To stay with the most basic
mtcars example possible:
heatmap(cor(mtcars), Rowv = NA, Colv = NA)
Created on 2021-12-19 by the reprex package (v2.0.1)
Note that it's the exact same data, these two heatmaps are identical in terms of what they show, but the first one looks nicer, and makes it obvious that there are 2 groups of metrics: the car weight/engine size/horsepower are correlated together, and anticorrelated with the miles per gallon etc. On the second heatmap it's not as obvious.
For the first argument, the important part is the column ordering, we wouldn't have something that looks much worse if we didn't represent the dendrogram (although I feel the presence of the dendrogram does signal to the reader that a hierarchical clustering was used, and the nice order is not a coincidence).
For the second part however, the dendrogram is important: sure, it looks like there are two groups, how convincing is it? The distances in the dendrogram help us see how well the groups are separated, and can become even more important when we have a more complex structure.