The Higgs field and the Higgs boson are interconnected but serve different purposes within the Standard Model. The Higgs field is a scalar field permeating all of space, proposed to provide particles with mass. As particles pass through this field, they interact with it; the more they interact, the more mass they acquire. On the other hand, the Higgs boson is a quantum or a particle manifestation of the oscillations in this field. It was the discovery of this particle that provided the experimental proof for the existence of the Higgs field.

Neutrinos hold a unique position in the Standard Model. Initially, they were thought to be massless, similar to photons. However, experiments have shown that they oscillate between different types (flavours), which is only possible if they possess mass. This discovery was surprising and posed a challenge to the original formulations of the Standard Model. Moreover, the exact masses of the neutrinos and the mechanisms that give them mass (whether it's the Higgs field or some other mechanism) are still active areas of research.

While the Standard Model is a powerful and successful theory, it does have its limitations. It doesn't include gravity, nor does it account for the mysterious dark matter and dark energy that make up about 95% of our universe. The model also doesn't explain why there's more matter than antimatter in the universe. Additionally, it relies on many parameters that must be put in 'by hand' or measured experimentally rather than being derived from the theory. Thus, while the Standard Model is an excellent description of the particles and forces we're familiar with, there's more to the universe than what it can currently explain.

Gravity remains a challenge in the realm of quantum theories. The Standard Model comprehensively describes three of the fundamental forces: electromagnetic, weak, and strong nuclear forces, using quantum mechanics. However, gravity is primarily described using Einstein's General Relativity, a classical theory. The two frameworks, quantum mechanics and general relativity, are fundamentally different, and attempts to combine them into a consistent quantum theory of gravity have yet to be successful. As such, while gravitons are theorised as force carriers for gravity, they haven't been incorporated into the Standard Model, and gravity remains separate from this model for now.

The discovery of the Higgs boson was a monumental validation of the Standard Model, but it also opened up new avenues of exploration in particle physics. It confirmed that the mechanism giving mass to elementary particles works as predicted. However, it also leads to questions about the properties of the Higgs boson and whether they match the Standard Model's predictions perfectly. Studying the Higgs in more detail can lead to potential new physics beyond the Standard Model. Additionally, understanding the Higgs might provide clues towards integrating gravity with the other forces, advancing towards a unified theory.