Motivation for local cohomology and local homotopy theories in Algebraic topology.

The local homology groups are defined as $H_k(U,U-x)$ where $U$ is a small neighborhood of $x$ in $X$. This is easily seen to be the same as the reduced homology group of the sphere when $X$ is a manifold. However let us look at real algebraic varieties.

For a space $Y$, we denote by $\overset{\circ}{c}(Y)$ the open cone over $Y$.

Theorem : Let $X$ be a real algebraic variety and $0 \in X$. There is a triangulated space $L_x$, a neighbourhood $U$ of $0$ in $X$, and an homeomorphism $f :U \cong \overset{\circ}{c}(L_x)$, sending $0$ on the vertex of the cone.

Corollary : $H_k(U,U-x)$ is simply $\widetilde{H}_k(L_x)$, the reduced homology groups of $L_x$.

Interesting example are given by the complex varieties $xy = 0, x^3 = y^2, xyz = 0$.

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For the second part of the question, I would say that there is no real difference since cohomology gives global invariants using local data. In a sense, everything is local. More precisely, all reasonable cohomology theory are computed by sheaf theory.

I want to complement Nicolas' answer. The local homology groups of a space can be useful even if we a-priori know that all the local homology groups are isomorphic. For example on a manifold we know that, $H^{n}(U,U\setminus \{x\})\cong H^{n-1}(S^{n-1})\cong\mathbb{Z}$. However, there is a choice made in this isomorphism, namely the choice of the generator of the homology of the sphere.

One can build a space by gluing copies of $H^{n-1}(U,U\setminus \{x\})$ over your manifold. A section of this space exists if and only if the space is orientable. A choice of section which is a generator at every point is a choice of orientation.

This allows one to speak of orientations of topological manifolds, which do not have a tangent bundle, because there is no smooth structure.