In both games you are playing a phylum of either mammals or lizards (250 million years ago or so). You place your creatures on an ever-changing board, seeking out biomes that (a) have food you can eat (including other creatures) and (b) give advantages to your genetic advantages (such as being amphibious, being fast, having armor, etc). You acquire these genetic advantages during the course of play (more later on that), and you can spin off new species that share some of the genetics of the parent species. The world is not a stable place, though, and the board changes frequently, generally becoming harsher (i.e. you need more of these genetic advantages to remain viable). Placement is governed by proximity to your existing creatures and size (bigger creatures can move farther). Tangentially, I note that in both games we have found the rules to be poorly-organized and somewhat confusing to learn.
In American Megafauna, genetic advantages ("DNA") come out one at a time and players bid on them. When you spin off a new species you inherit all of the DNA of the parent (I think). You have 14 counters for each of four species, and the game supports four players (five with the expansion). So the board can get fairly crowded. DNA is tracked by little cardboard tents on a playing mat, which are pretty hard to see across the table. The winner is the player with the most creatures on the board when the game ends.
Bios Megafauna makes several changes:
The bidding mechanism for DNA is completely revamped. Several cards are dealt out on the table; the cards represent a mix of DNA and opportunities to start new species. A player can take the first card for free, pay one for the second, pay two for the third, and so on. If you take a card with payment tokens on it, you get them. Tokens are quite limited. Cards are replenished, coming in at the end.
There are two ways to spin off new creatures. One is similar to the earlier game, but you choose which traits to inherit and indicate them with a limited pool of tokens. So, for example, you may have one "nocturnal" chit that you can use for this, and once it's gone it's gone. The other way is with cards. A species card will indicate several genes, and to play it you must match at least half of them with an existing creature (which will be the parent). So if you have a creature that's marine, nocturnal, and doubly speedy, and the card has marine, speed, armor, and insectivore, you can use it. The new creature has the DNA on the card. New creatures can therefore have DNA that you didn't previously have represented, which is handy.
Presumably because of the connection to other games (which use the same idea), Bios Megafauna adds non-DNA characteristics that you can also acquire through cards (things like manual dexterity, language, and a couple others I'm forgetting). These can give some additional advantages, but in our games they haven't really been well-integrated.
When tiles (mostly biomes) are removed from the board they go into a pile that will eventually turn into victory points. Points are distributed multiple times during the game based on population at that time -- so instead of "biggest at the end wins", you have several checks and players' fortunes can ebb and flow.
There are many fewer creatures in play. You get eight tokens (wooden counters now rather than cardboard tents) per species (still four of those), and you have to spend some of them on "tracks" -- one to indicate your current size, and one for each of the "roadrunner" DNA (nocturnal, speed, armor, marine). So yeah, go ahead and acquire those traits, but you do it at the cost of board position. In our games it has not been uncommon to have only one or two tokens of a species on the board. (In both games, if you go to zero that species is extinct -- flush the cards, reclaim your tokens, and carry on.) The game supports up to four players.
Regarding physical aspects, the biome tiles are larger (so easier to see) and the annoying little tents are gone. However, DNA for any given species is now represented by the combination of counters on tracks, DNA on cards, and the chits used in spinning off creatures. Add to this that the counters for the different species are not that visually distinct from a distance, and we have found that it's just as hard to see what that tasty morsel you're considering eating has in place as it used to be, but it's differently hard. In both games players are constantly having to ask each other questions like "how speedy is that guy again?" and "does that have armor?".
Anecdotally, I have heard that "total party kills" -- that is, world-destroying events that end the game -- are much less likely in the new game, and that this was one motivating factor. Beyond that I don't know why the author reimplemented the game.
The new game is designed to fit into a family of games in ways that aren't entirely clear to me. We have played one of these, Origins: How We Became Human, once; it shares the non-DNA traits of Bios Megafauna but the rules are very different. As the name implies, it is set later in time. I haven't played it enough to make further connections.
How to choose between American and Bios? Both have merits and both can make play difficult in different ways. It seems to be easier to recover from misfortune in Bios and the cards provide more options; American is a harder and I think harsher game, which may be what you want sometimes.