Note that this has nothing at all to do with residuals as such. It applies generally to looking at any distributions.
The two graphs do not have exactly the same purpose. Be clear that a symmetry plot checks for symmetry or asymmetry and would look simple for many symmetric distributions that were not Gaussian, e.g. t distributions with finite degrees of freedom. But there is still a question of whether the graphs contradict each other.
I here assume familiarity with normal probability plots (historically often so named, although Gaussian quantile-quantile plots is a minority preferred name). See for example this explanation.
However, symmetry plots seem less used and bear some explanation.
symplot, as the axis titles imply, pairs values above and below the median and plots (largest $-$ median) vs (median $-$ smallest), (second largest $-$ median) vs (median $-$ second smallest), etc. and the reference line is thus (value in upper half $-$ median) $=$ (median $-$ value in lower half), implying symmetry of distribution.
What you can't tell easily from
symplot in cases like this is how many values are in the middle, often approximately symmetric part of the distribution and how many in the rest.
It is easy for
symplot therefore to impart a pessimistic message because points may be heavily overplotted near the middle of the distribution.
Here is another example. I simulate 95% of values from a Gaussian and 5% of values from a gamma with the same variance (but evidently different skew).
This is the Stata recipe used:
set obs 10000
set seed 2803
gen y = cond(_n <= 9500, rnormal(6,10), rgamma(1,10))
symplot seems to flag lack of symmetry (and thus lack of normality) more prominently than the normal probability plot (Gaussian quantile-quantile plot) flags lack of Gaussianity.
It's manifestly the same data, but the tail is inevitably more prominent in one graph than another. In addition to the question of overplotting, in a symmetry plot all the bad news is usually lumped together at one end; in a normal probability plot there is often bad news in both tails.