They basically mean the same thing, though "grave" has more impact (but not too much more.) It is also used in a more negative situation. Also, the usage is different. "We need to take a serious look into the problem." = We must make a good effort to look into the problem. You would not use "grave" "When the student was brought in to the office for causing trouble, the look on the principal's face was grave." You could use "serious" here, but, but grave implies to a greater degree that something bad is about to happen or has happened already.

You can use both as they are synonyms