THE AMBUSH
The way the opposition (Zeon in 0079, ZAFT in Seed) secretly invaded the respective space colonies (Side 7 in 0079, Heliopolis in Seed) in the first episode, are the same. They hid in atop a cliff with trees surrounding them. Both Zeon and ZAFT soldiers used binoculars to watch a hidden military base from afar. If you've seen either series, you'll know where this location is.
WHITE BASE vs. ARCHANGEL
Both are space battleships used to help the main "hero" Gundams for maintenance. In 0079, it's known as White Base (and known to the Zeon forces as The Trojan Horse). It houses the RX-78 Gundam. In Seed, it's known as Archangel (known to ZAFT as the Legged Ship) and it houses the Strike Gundam. The comparisons between them are uncanny. The Archangel is more rounded out while the original White Base is more angular.
White Base, Archangel
MAKING IT TO EARTH
Both White Base and Archangel made their trip towards Earth after their run from the opposition from their respective space colonies (mentioned in first comparison). Not surprisingly, both trekked through the desert upon landing.
YOU KNOW WE'RE BAD IF WE GOT "Z"
Have you noticed that Earth's enemy forces start with Z's or just contain them somehow? In 0079, the enemy forces are known as the The Principality of Zeon. In Seed, the enemy forces are known as ZAFT (Zodiac Alliance of Freedom Treaty). In Gundam Wing, they're simply known as OZ.
NEW TYPES vs. COORDINATORS
Both are genetically-enhanced people. They tend to do things better than "normal" people. New Type people are found in 0079, while Coordinators are found in Seed. They oppose the regular people (also known as Naturals in Seed) of Earth since many oppose the idea of genetic enhancements (maybe against religion?). Main characters Amuro Ray (from 0079) is a New-Type, and Kira Yamato (from Seed) is a Coordinator ... incidentally, both fight for Earth.
MASKED MEN
Both series feature a character that uses a mask to hide their identity. These people are mysterious, brooding, pretty boys. Even Gundam Wing featured a character with a mask, who went by the name of Zechs Merquise (later Milliardo Peacecraft). In 0079, it's worn by Char Aznable. In Seed, it's worn by Raww Le Klueze, a ZAFT commander. The only difference here is that we never get to see Klueze's face ... just yet.
Char Aznable, Raww Le Klueze, Zechs Merquise
360° ROTATION
In both series, the main battleship had to rotate 360° horizontally for some reason. In 0079, it was flipped to shake off a mobile suit (which was Ramba Ral's mobile suit known as the Gouf) that was clinging on White Base. In Seed, it was flipped to shoot at a aquatic mobile suit that was directly under the base underwater...since the Archangel was travelling by sea on Earth. There was only one aquatic mobile suit at that time, and it was giving Archangel a hard time.
HARO, GENKI! HARO, GENKI ... NAI!
In both series, there are Haros. Haro is basically a pet robot. It can use its arms and legs to walk or climb but more often it'll bounce along like a basketball or flap its "ears" to hover momentarily in the air. In 0079, it was created by Amuro Ray. In Seed, they were created by Athrun Zala. Athrun's creations does not have arms and legs, but rather bounce to make it from A to B. All of Athrun's Haros were given to his fiancée, Lacus Clyne.
0079's Haro, Seed's Lacus Clyne with Haro
MARRIAGES
There were arranged marriages in both series that weren't meant to be. In 0079, it was between Mirai Yashima and Cameron Bloom. They broke up on space colony Side 6 because Cameron was a coward in the war while Maria had to fight for survival in White Base. In Seed, it was between Fllay Allster and Ssigh Argyle. They broke up because Fllay wanted to be closer to Kira so he can die ... don't ask me why ... crazy b**** ... ^_^
Mirai Yashima, Cameron Bloom, Fllay Allster, Ssigh Argyle
IT'S A CYCLE ... FIRST SPACE, THEN EARTH, THEN SPACE AGAIN
In 0079, White Base had to leave Earth to serve as a decoy for the Federation forces in space. In Seed, the Archangel had to leave Earth because The Kingdom of Orb wanted to self-destruct rather than the Earth Alliance or ZAFT take control of it. To do this, they had to send the "seeds of hope" out before they can do that. I've been told that this cycle has also been done in Gundam Z and ZZ Gundam.