Suleiman is the consummate operator, and has strong diplomatic ties to pretty much the entirety of the West, Israel, PA, Jordan, Yemen and Saudi Arabia.
He is an important part of the military; actually the most active part for the last 30 years- since Egypt has been heavily modernized and the Egyptian Army has no 'operational military action' (with the notable exception of Gulf War I,) because the Suez is considered a DMZ. The navy has conducted exercises against Somali piracy, and with the U.S. navy, but the Egyptian army remains economically focused:
The Egyptian military through patronage controls roughly 35% of the countries GDP, largely through manufacturing. If Egypt is Walmart, the Egyptian military is China.
Tantawi is a higher ranking officer, and notably a more charismatic and pol-ish (ed) figure. He is tightly involved with the U.S. military, and frequently is in Washington or speaking with Gates. Its reported that when Gates wanted Egyptian anti-terror support in Yemen then he spoke with both Suleiman AND Tantawi to get their support.
There is a lot of jubilation and consideration of this being a complete break with the previous regime, but that would be a complete misreading: the Mubarak regime consisted of a coalition with Mubarak and the patronage he supported, and the military and their patronage. The plan was for Gamal, Hosni's son to become heir apparent in Sept, but that plan failed.
As soon as this uprising started around the 28th Jan, I thought to myself "there will be a coup d'etat", and the last day was a brilliantly presented coup d'etat: The military sides with the protesters in a communique, saying "Mubarak" is leaving. Hosni predictably says, "no I'm not." Suleiman then gives Mubarak the final shove and the military, Suleiman included, take over.
The military has a year to position themselves towards a new regime, and they largely control the economy.
In an Egypt where the basic source of dissatisfaction is economic disparity, the people are about to see how entrenched 30 years of military patronage can get. This isn't one guy that you can just push out. This is THOUSANDS of guys who quite literally own the lions share of manufacturing in Egypt, and who have zero reason to give up that ownership.
Its a nice idea that Egypt can become highly democratically driven, but there are realities involved: if they stop shipping NG to Israel, Israel will develop their own recently discovered NG fields in their territorial waters and the Egyptian people stop profiting from the sale (although I think what ultimately happens there is the contracts are renegotiated to a market price; less beneficial to Israel but the NG still flows)
The 80 million in Egypt require food, and as their agricultural environment inevitably degrades, and China's, the largest consumers, wheat crop crashes, and they will very soon realize the folly of alienating large trading partners in the U.S. and Israel, by being as (democratically) nationalistic as many would wish, because its going to have a highly detrimental effect economically, and you can't eat pride.