DD had Babybug until 23 months, DS until 3 and I think that DS keeping it longer was a better choice. If you have a subscription to Babybug, you can move up to Ladybug at anytime--- you just contact them and tell them.
Animal Baby is Your Big Backyard's younger sibling and published in the same format as Babybug (almost a board book, but more flexible).
Personally, I like Babybug better than Animal Baby because it covers more subjects. Additionally, I really like that they have a "Kim & Carrots" (a toddler and stuffed bunny) story every month--- it is something a toddler can begin to predict and look forward to.
That said, if your child is really into animals maybe Animal Baby would be a better choice. If you can, check if your library has copies. Either one is GREAT and they also are relevant for years (unlike news magazines, or ones on popular culture). They also hold up very well--- we have them from the last 5 years (well, DS actually moved to Click this last November so we aren't getting them anymore) and most of them are still really nice.