You don't get to choose. Processors are (with a very small number of exceptions) fixed in their endianness, so when you have data to manipulate, you MUST put it into the proper format for your CPU. It's not a choice, and it's not a performance issue: it's just what must be done.
The only time it comes up much is when serializing data. Many common network protocols were designed with big-endian on-the-wire representations. If you read those bytes straight into memory on an Intel CPU, you have to swap the bytes around to make them little-endian before you can work with them. This isn't a big deal, and again, isn't a choice - the network protocol is what it is, and you don't get a choice about how to put things on the wire if you want other systems to be able to understand your packets.