Maximally complete field does not have immediate extension. This gives it better chance of having its algebraic properties completely determined by the valuation group and the residue field. This is still conjectural however, and is a direction of research; it's known to work in characteristic 0, but you don't need the strong property of maximally completeness, only Henselian is enough. If this works, it potentially gives us an algorithm to resolve all questions about the field by reducing to residue field and valuation group, which are much easier to answer.
I'm not aware of any applications of the concept of spherically complete fields in general, but particular examples - the p-adic numbers - are very important in number theory, and part of what make them important is their compactness, which is the property of spherical completeness.