The main difference I see, is that spiritually dead means you're separated from God. And, you don't really care, or don't really notice. Spiritually dry is exactly that... a dry spell. A temporary drought where there is little to no passion for Christ.
"For you will be like an oak whose leaf fades away or as a garden that has no water." Isaiah 1:30
I find it rather interesting that the signs of being spiritually dead or spiritually dry are quite similar:
1. We do things because we are supposed to, not because we want to. We give to others out of obligation rather than desire.
2. Sermons, scripture reading, hymns - they're all given, listened to, sung with a lack of passion for the words.
3. There's a lack of passion for Christ. Our faith is supposed to be contagious, but when there is no passion behind that faith, it is not contagious. People aren't motivated to move towards Christ because we have no motivation to show them the joy God brings us.
I'm probably missing a few signs here, but I think these 3 sum it up pretty well.
So, what causes us to become spiritually dead or dry? To a degree, I think some of what I am about to list can be causes of spiritual death or drought...
... These three block us from being able to see the grace of our Lord, and keep us from feeling the joy that grace can bring us. Depression makes us unable to see the joy; worry drains us emotionally, and eventually spiritually, making us weak; and anger makes our hearts hard, and keeps us from giving and receiving love from anyone - including God. Each of these can bring about a dry spell, which, if left untreated for long enough, can bring on spiritual death.
... Yes, we are all sinners. But some of us reside within sin so much, and are unaware or uncaring. It becomes so much of what we seek, that we become separated from God. One of the biggest sins (in my opinion) which can cause this separation is pride. We get so wrapped up in what we want and how we want it - in how we know it should be - that we lose sight of God and His will.
I saw one really good article on how to know if you're spiritually dead, and it talked about the Church - how those who are spiritually dead don't care if their church also dies. They'd rather fire a pastor who was bringing about change and growth, than allow themselves and the church to continue growing. They'd rather see the church die. This is the type of pride I am talking about. (5 Signs You're Spiritually Dead)
How do we move forward if we find ourselves in a dry spell?
"So let us know, let us press on to know the Lord. His going forth is as certain as the dawn; And He will come to us, like the rain, like the spring rain watering the earth." Hosea 6:31. Give your worries to God. Place them at his feet.
3. Let go of anger. Forgive whomever you're angry with.
I realize these are all so simple - too simple. And they aren't as easy as they sound. But we must "press on to know the Lord."