You use alternately to say that two actions or processes keep happening regularly after each other.
1. Each piece of material is washed alternately in soft water and coconut oil.
2. The little girl had alternately sulked and made scenes.
You use alternatively to give a different explanation from one that has just been mentioned, or to suggest a different course of action.
3. Or alternatively was he short of cash because he had never been to the Rosses’ house at all?
4. Alternatively, change seats at once.