What is the difference between reliance on miners on Drivechains and Lightning?
I would consider these two crucial differences, so I guess you answered your own question. I guess the more important distinctions, though, are about the systemic impact of both:

Firstly, Lightning can simply be done with bitcoin as it is (OK, we needed segwit, but we needed that anyway). So it would be hard to keep people from using Lightning. 

Secondly, the risk of miners stealing bitcoins in LN channels is probably not a problem: LN deals with small amounts, miners would have to collude with one side of every channel they want to steal, and breaking Bitcoin's censorship resistance in order to steal some LN funds would be such a nuclear attack that it would probably tank price AND lead to a PoW change. 

On the other hand, IIRC drivechains explicitly expect miner consortiums to steal funds in unpopular sidechains. Very different risk profiles IMHO.