The concept of "practice is enlightenment" is not specifically Soto. The idea that the original nature is enlightened and all deeds are the manifestation of that is generally Zen. The important point is the view of "buddha-nature" which is shared by both Dzogchen and Zen. But if we compare Dilgo Khyentse's description with a modern Zen intro to practice we can find how many of those guides include regular formal practice and other things. But it can be very similar too - a lot depends on who you ask.
Let's look at what Anzan Hoshin says
, "Zen is entering into things as they are, beyond concept and cosmology, beyond separation and duality, beyond personality, and into the intimacy and richness of this whole moment. It is a radical questioning into whatever arises as our experiences and true entry into the nature of experiencing. Zen is the day to day and moment to moment practice of this moment."
But if you actually ask how to do that we find something like this
: "As well as sitting, we have walking practices called kinhin; we have movement practices called kata; for eating we have something called oryoki practice. There are also sleeping practices, dreaming practices, because in Zen we want to work with each aspect of our experience, completely. So then, none of these things are themselves Zen. Sitting is not Zen, kinhin is not Zen, oryoki is not Zen. The entire continuum of the training is Zen. ... You are not going to learn Zen in a week or a year or fifteen years."
Zen - in its rhetoric - is rarely so carefree as they like to present Dzogchen. On the other hand, Dzogchen doesn't stand alone but rather as a part of the nine vehicle training of the Nyingma school. Comparison then is very difficult. Especially because when it comes to the practical matters Zen doesn't say you're enlightened just like that, so simply be open and rest. Although "just sitting" sounds like that, but then, why only sit, or why sit at all?
Another important difference is how actual explanation and pointing out of the nature of mind is so hard to find in the Zen teachings while in Dzogchen it's right there. In Zen they like to say you have to do this and that and then you may find the nature of mind. In Dzogchen they try to show it right at the very beginning and then keep working with that. The way Dogen teaches is quite far from how Longchenpa elucidates the meaning. To read Dogen one needs to be able to decode the whole thing, and even then it's hard to gain much from it. Longchenpa is a lot more "user friendly".
"While teachers of the middle way, mind only, transcendent wisdom, mantra, and other schools may have their own assertions, the fulfillment of those intentions is the same. There is not a single thing that is not contained within mind."
(Gampopa to Düsum Khyenpa, in "The First Karmapa", KTD Pub, p254)
“If you recognize the world of appearance and existence as the mind, realize the mind itself as empty, and have no grasping at the superiority of your realizations — this is the ultimate view."
(Chegom Dzongpa, in "The Book of Kadam", Wisdom Pub, p609)