Batman Adventures #1 -- No Asylum
Batman: Gotham Adventures #1 -- With a Price on His Head
Based on these two issues I really prefer Adventures over Gotham Adventures. In part it comes down to the character designs. In particular I find the look of the Joker in Gotham Adventures to be jarring. He doesn't look human in that series. The Joker may be twisted but he's a twisted human being.
The addition of Robin, Batgirl, and Nightwing in Gotham Adventures doesn't add much although it certainly changes the dynamic of the series. In some ways the story feels cluttered with the addition of Batman's three protégés. There isn't enough good character interactions among them to justify including them all in this story.
Finally, I'm not crazy about the story in the first issue of Gotham Adventures, whereas I was riveted by the one in Adventures. The one in Gotham Adventures seemed a little too straightforward and formulaic. The one in Adventures begins with a mystery (Penguin is mayor and Batman is a fugitive) and then heaps more mysteries on top of that one. I also liked the character interactions in Adventures better, especially among the inmates of Arkham Asylum.
Despite having fewer pages Adventures also includes a backup story that recaps Batman's origin quite succinctly.
Both of these comics were available for free on Comixology. Both issues were written, penciled, inked and colored by the same team. I'm considering buying more of Batman Adventures. They cost $0.99 each.