Main Difference
The main difference between Topology and Topography is that the Topology is a subfield of mathematics and Topography is a field of science.

Topology
In mathematics, topology (from the Greek τόπος, place, and λόγος, study) is concerned with the properties of space that are preserved under continuous deformations, such as stretching, twisting, crumpling and bending, but not tearing or gluing.
An ndimensional topological space is a space (not necessarily Euclidean) with certain properties of connectedness and compactness.The space may be continuous (like all points on a rubber sheet), or discrete (like the set of integers). It can be open (like the set of points inside a circle) or closed (like the set of points inside a circle, together with the points on the circle).
Topology developed as a field of study out of geometry and set theory, through analysis of concepts such as space, dimension, and transformation. Such ideas go back to Gottfried Leibniz, who in the 17th century envisioned the geometria situs (GreekLatin for “geometry of place”) and analysis situs (GreekLatin for “picking apart of place”). Leonhard Euler’s Seven Bridges of Königsberg Problem and Polyhedron Formula are arguably the field’s first theorems. The term topology was introduced by Johann Benedict Listing in the 19th century, although it was not until the first decades of the 20th century that the idea of a topological space was developed. By the middle of the 20th century, topology had become a major branch of mathematics.

Topography
Topography is the study of the shape and features of land surfaces. The topography of an area could refer to the surface shapes and features themselves, or a description (especially their depiction in maps).
Topography is a field of geoscience and planetary science and is concerned with local detail in general, including not only relief but also natural and artificial features, and even local history and culture. This meaning is less common in the United States, where topographic maps with elevation contours have made “topography” synonymous with relief.
Topography in a narrow sense involves the recording of relief or terrain, the threedimensional quality of the surface, and the identification of specific landforms. This is also known as geomorphometry. In modern usage, this involves generation of elevation data in digital form (DEM). It is often considered to include the graphic representation of the landform on a map by a variety of techniques, including contour lines, hypsometric tints, and relief shading.

Topology (noun)
A branch of mathematics studying those properties of a geometric figure or solid that are not changed by stretching, bending and similar homeomorphisms.

Topology (noun)
A collection τ of subsets of a set X such that the empty set and X are both members of τ and τ is closed under finitary intersections and arbitrary unions.

Topology (noun)
The anatomical structure of part of the body.

Topology (noun)
The arrangement of nodes in a communications network.

Topology (noun)
The properties of a particular technological embodiment that are not affected by differences in the physical layout or form of its application.

Topology (noun)
The topographical study of geographic locations or given places in relation to their history.

Topology (noun)
The art of, or method for, assisting the memory by associating the thing or subject to be remembered with some place.

Topography (noun)
A precise description of a place.

Topography (noun)
A detailed graphic representation of the surface features of a place or object.

Topography (noun)
The features themselves; terrain.

Topography (noun)
The surveying of the features.

Topology (noun)
the study of geometrical properties and spatial relations unaffected by the continuous change of shape or size of figures.

Topology (noun)
a family of open subsets of an abstract space such that the union and the intersection of any two of them are members of the family, and which includes the space itself and the empty set.

Topology (noun)
the way in which constituent parts are interrelated or arranged
“the topology of a computer network”

Topography (noun)
the arrangement of the natural and artificial physical features of an area
“the topography of the island”

Topography (noun)
a detailed description or representation on a map of the physical features of an area.

Topography (noun)
the distribution of parts or features on the surface of or within an organ or organism.