Strategies should provide direction (why and how something should happen). Plans should include actions and cover who, what and when.
Whichever one you're working on, people are bound to request more practical actions in your strategies, and more long term vision in your plans!
Another challenge for local government is the linkage between strategies and plans, especially with the amended Local Government Act 2002 requiring 30 year infrastructure strategies.
In theory infrastructure strategies (first required in 2015) should be guiding activity management plans and regional land transport plans, but it will take some time for these new documents to really make a strategic difference.