Things I've 100% noticed from the new SMN (with a bit of speculation too).

[Gallus]

1 - Demi Bahamut seems to no longer cast wyrmwave on OGCDs, notice how he does his usual wyrmwave when the SMN casts ruins but he doesn't do anything when the SMN casts tri-disaster. This obviously simplifies things, as you don't need to save your OGCDs or time them up with your Bahamut phase to make optimal damage. Also, no more Virus for that extra wyrmwave. I actually sort of like that.

2 - Pretty sure the skill aetherflow is entirely gone on the SMN. Painflare and fester still cost 1 stack, but now Energy Drain is your new aetherflow. Notice how in the video the SMN casts Energy Drain and they get their 2 stacks back.
I suspect Energy Drain will have a CD now.

3 - As for phoenix itself, there's something very weird going on in there. Unlike with Bahamut, the SMN himself doesn't seem to cast anything but rather command Phoenix to do stuff instead. The first thing casted seems to be your spammable exclusive Phoenix ruin (which seems to be an instant cast), the second thing casted seems to be a more potent spell (likely to have a short CD allowing 2 or 3 of these ones during your phoenix phases, no idea if it's gonna be a CD or OGCD though), the third one is the equivalent of Bahamut's Akh Morn, likely to be OGCD and be able to be casted once or at best twice during your entire Phoenix Phase. So we might have an entire different set of buttons for the Phoenix transition (likely no more than 3 or 4 though). However, this is speculation.

4 - Dreadwyrm Trance? Shadow Flare? Aetherpact? Deathflare? Enkindle? Sustain? Rouse? Pretty sure all these are gone.

5 - Your rotation seems to consist on going between Phoenix and Bahamut back and forth.

6 - This is just speculation, but from what I've seen, the Phoenix + Bahamut uptime could be almost 100%. Why do I draw this conclusion? Because pet actions have been eliminated, now Garuda and Ifrit only have one action, so having them there just plain autoattacking doesn't sound very exciting, and if my predictions are correct and rouse is gone too this further proves I might be right about this, since now, there's no reason to wait to hold on Bahamut/Phoenix because of your Roused Egi. So perhaps you may want to have Garuda/Ifrit for a few seconds out to cast their one and only action now, and then swap them for Bahamut/Phoenix. This is speculation though.

7 - I noticed one new spell too in the Bahamut Phase. Could very well be a Ruin V. It seems to be hardcasted (and it seems to be a relatively long cast). (I could be totally wrong here and as another posted mentioned, it could very well be tribind, my apologies).

8 - I predict now Bahamut and Phoenix cost being either 2 stacks of aetherflow or 4, depending on Energy's drain new CD. The 2 feathers on the Phoenix icon lead me to think that. Each feather either costs 1 or 2 stacks. I'm more inclined to think each feather costs one stack. Speculation again though.

9 - Both Summon Phoenix and Summon Bahamut might be locked behind a 25-30 sec CD, to prevent spamming one.

10 - As for Titan-Egi. I predict it'll become your defensive pet. It won't transition into a greater primal but it might give the SMN a more defensive approach where Titan-Egi keeps debuffing the target he's attacking so that it does a bit less damage or Titan Egi's action casting an AOE defensive buff on your partners. Titan Egi's new action is called Earthen Armor so everything seems to be pointing this way. This of course will come at the price of doing less damage.