I have no idea what the difference between the two is because I am not involved in mining. I directly invest in coins, but I'm curious about the differences between the two in case I decide to start mining.
In theory the difference should be that in case of cloud mining you buy (any) hash rate (which you may also able to resell to other users if you no longer want to mine), while for hosted mining you buy actual miners and that may not be so easy (availability, more expensive, ...). Especially in case of hosted mining you'll have to pay for all the expenses, from electricity to maintenance. And you also don't get your hands on the machines.
In reality cloud mining means websites that have no miners, display some nice numbers and offers to attract unsuspecting newbies into investing and they don't pay (or even more, they ask for more money pretending you need to pay in order to get your money). Plain scams.
In reality hosted mining is also dangerous since you cannot be 100% certain the host will keep his promise. I've seen stories with the host simply making some day all the miners mine to his own wallet and not answering to calls any longer. Of course, if you have a good contract and the identity of the host was real, you can start (a costly) lawsuit.
As @mocacinno said: it's better to avoid both (unless you have
very reliable information you can react legally if they attempt to scam you, information most won't give, or not a real one).