Answer: I would not expect your tree to die. An exception would be if the main trunk broke or if a large portion of the foliage is removed or, on a grafted tree, if the tree was damaged below the graft.
When a branch is broken in a storm, it is a good idea to trim away the damaged part making a clean cut with a sharp hand saw or possibly a bypass pruner if the branch is small. This helps the tree heal over the damage and prevents further tearing of the bark.
Your cut would be made just next to and outside of (beyond) a remaining branch that heads in the direction you would like the tree to grow. When you cut, take care not to leave a stub, but at the same time you also want to cut just outside the branch collar, a slight swelling that surrounds the base of the branch. (The branch collar is what will grow and heal over the cut.)
Since you are seeing new growth, you may be able to cut just outside one of the new twigs and allow it to replace the missing branch. If there are numerous small stems starting there, you may also want to simply rub off most of them to allow one to become dominant.
Japanese maples actually adapt fairly well to pruning so I would not expect the storm damage to kill the tree. Removing a branch would usually stimulate additional new growth. What you are seeing now is new growth trying to grow to replace the damaged portion, this would be a normal response.
If the tree is a valuable tree or important in your landscape you may want to look at a book or two about pruning before you begin. You might also consider consulting with a professionally trained and certified arborist who specializes in ornamental trees such as this. Good luck with your tree.
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