Disclosure, I'm the CEO of Eris Industries
The major difference between the Consensys offering and the Eris Industries offerings on Azure are two fold.
At Eris Industries we are focused as @5chdn rightly noted on being a blockchain-backed application platform. In that regard with eris you can one click install a variety of different blockchains (including geth, btcd, bitcoinclassic, zcash (soon) and others), services required for your application (docker image watchers, log rotators, etc.), have easy access to IPFS, and a range of other tools and services required to build blockchain-backed applications. The Consensys offering provides a one click install to their quite nice contracts development tooling which is focused one blockchain client: geth (or a network of geth clients of course).
Included in eris is the capability of using eris chains which is a blockchain client that currently utilizes an ethereum virtual machine, a tendermint proof of stake consensus engine, and a permission layer. The major differences between eris chains and geth are that (a) when geth is ran in private chain mode it is still running its proof of work algo (sometimes this is desired and sometimes not) whereas eris chains utilize tendermint's proof of stake algo; and (b) geth lacks a permission layer (meaning if an attacker were to get within the network it would still be able to attack the chain).
One can run private geth chains with eris just fine as well; from our perspective folks should use whatever tooling helps them get the job done.
The similarities between eris chains and geth, though, are that contracts do and should run the same on both geth and eris chains. We use slightly different API/RPC structures but at the contracts layer everything should operate in the same manner.