In The Fellowship of the Ring, Gandalf describes Strider hunting down Gollum which gives Gandalf the information about Gollum's past (how he found the Ring, that he was captured and escaped Mordor). Besides the cameos (Arwen, wood elves, Nazgul, Bree). Is there any other continuity?

I found three breaks in continuity.

1In LOTR, both Gandalf and Aragorn hunt Gollum.
2THFG has Gandalf send Frodo to Bree to meet Strider, but in LOTR Gandalf means to meet Frodo himself and Strider is the backup plan.
3Perhaps most importantly, when Gandalf met with Bilbo at his bday party, he didn't not know it was The Ring of Power, only suspicions. Only when Gandalf throws it into the fire is he sure and then immediately tells Frodo Gollum's story. In THFG, Gandalf already knows Frodo has the Ring. If Gandalf had known, he would not have let the Ring stay unguarded, especially not telling Frodo its importance.

Anything else?

Well Gandalf and Aragorn hunted Gollum together just at the beggining, finally Gundam gave up but Aragorn did not so he hunted Gollum and taked him to the elves of the forest.

Still, in beginning of THFG Galdalf seems to be aware of AND describe the One Ring when talking to Aragorn:
"...the enemy does not yet know that the ring has been found..."
Theoretically THFG can take place after he sent Frodo to Brie but I'd say its too short of time for all events in THFG because Aragorn has a date with Frodo in Brie...
I say that THFG storyline not fit in 100% in LOTR - this fact is a bit annoying but still its a FANTASTIC fan-art masterpiece. Camera, music, etc adds perfectly to the best triology ever made. I really enjoyed THFG!
(Posted same text on IMDB, I really wish for more opinions on storyline)

Okay, this isn't really a break in continuity-- but:
To me it doesn't seem that Aragorn would introduce himself to another ranger as 'Strider'; I have the impression from the books that this is a name given to him by people who don't really know him or respect him (like the hobbits who live in the Shire) and that it is just adopted by Frodo, etc. because that is how he introduces himself to them. Likewise, I don't really think another ranger would respond by being all "Ah, I've heard of you". The name Aragorn might produce such a response, though. ..Not positive, just something that strikes me.
(Also, does Aragorn ever address Gandalf directly as 'Gandalf' in the books or other movies?)

Here I have to agree, as Aragorn is chief of the Dunadan, and only known as Strider to the folk of Eriador who the Dunadan as Rangers are protecting, he would be known to all the Dunadan