In the context of phone apps, is there a difference between these words? How about in other contexts?
It depends on the context you're using these words. If you say something is "费电", take it as adj., then it mainly means the device consumed a lot of electricity. Sometimes, if you take it as a verb, then it means the device is consuming (usually, a lot of) electricity. So from these, you may see "费电" is commonly derogatory. While "耗电" is just a neutral word, which is used to describe the power consumption. For example, this device "耗电" 1kw. This is a just neutral description. However, if you use an adjective word, like "很(pretty much)", before "耗电", then it also become a derogatory word which means it consumes a lot of power. So if you're try to pick a word in phone app, in most of case, I suggest "耗电". For example, tap the button of "耗电排行(Power Ranking)", to show a list of apps with their power consumption statistics.
In the context of phone apps, 这个app很费电 vs 这个app很耗电: there is a subtle difference.
这个app很费电 sounds that the App consumes more power than you think it's necessary. It can be a way of complaining.
这个app很耗电 is neutral. It sounds that the App consume lots of power, which is necessary by its nature. For example, a camera App is generally considered as more power-consuming App. You will accept its consumption because it's the common sense.
In practice, people might not really care about the nuance and sometimes they use them interchangeably.
I think that nuance applies to other contexts as well. For example, we say: 空调很耗电的, it's the common sense that AC consumes more power than other household appliances, so you wouldn't complain about it. On the other hand, 这空调很费电的 sounds like this AC consumes more power than any other ACs(consume more than it's necessary). The latter sentence can be a complaining.
Hope this would help you.