Can anyone tell me why someone would want their smart contracts running on Counterparty instead of Ethereum? Is there any advantage besides the security of the Bitcoin blockchain vs the security of the Ethereum blockchain?
No, apart from the fact that your money or assets are encoded on the only chain with real consensus and a workable solution to the Byzantine General's problem, there is no benefit. I guess if you don't care about network stability or who overwrites your coins and assets, you are free to use Ethereum.
(And Counterparty smart contracts will let you read BTC blocks, BTC transactions, BTC balances, and engage in nlocktime and other BTC scripting)
As to whether Counterparty will become used for a specific project, I can't say, simply because I don't know.

It really just depends on what engagements materialize, and the requirements of each one.
The way I see it: Once you establish rapport, giving tangible metrics of certain inefficiencies (in applicable fields) resolved by using Counterparty instead may be a stepping stone once a private solution is implemented. However, only calculated graphs, simulations and a 3rd party assessment will be convincing. I do hope these solutions are by definition modular and interconnectable as stated before.