I put this as an answer because it's too long for a comment. It will probably also be quite rambly - feel free to edit and clean it up.
If the underlying is dependant on the vol in a non linear fashion, then you need to get the distribution correct in order to correctly price the product. When the LV surface is created, it is calibrated to products that do have vol of vol exposures, but the lack of the LV model's ability to accurately model these means that any difference is projected onto the local vol surface.
The reality is that the LV model is not correct, it is an approximation. If we calibrate LV to vanilla options, then we can correctly price vanila options. We can also correctly price anything that can be approximated by a linear combination of vanilla options. If you now take a different class of derivative, and try and price that, then we are effectively extrapolating. And with that, there will likely be some error. This is a projection error. If our model is appropriate for the derivatives we are pricing, then the projection error will be small (and we can consider what we are doing more an interpolation than an extrapolation). If we try to price something that depends on something we're not actually modelling (i.e. vol of vol) then we shouldn't be surprised if we have a larger error.
Another reality is that SLV is not correct either, but what it does do is give us flexibility to calibrate a dynamic for the vol of vol. This allows us to reduce the projection error when we're dealing with products that have this dependency - but we need to calibrate to products that do have this dependency, otherwise we will just get meaningless parameters (i.e. you can't calibrate a model to predict whether or not your wife is angry at you to if your only input is the phase of the moon...).
As for why autocallables price differently under LV and SLV, it's because of vol of vol yes, but this is mainly because a side effect of that is the forward vol / conditional vol dynamics. LV models give very flat forward skew, while SLV allow a fwd skew more easily. Because of this, things like options conditional on not knocking out in one year at 100% price differently under the two models.