Main Difference
The main difference between Diameter and Width is that the Diameter is a straight line segment that passes through the center of a circle and Width is a measured dimension of an object.

Diameter
In geometry, a diameter of a circle is any straight line segment that passes through the center of the circle and whose endpoints lie on the circle. It can also be defined as the longest chord of the circle. Both definitions are also valid for the diameter of a sphere.
In more modern usage, the length of a diameter is also called the diameter. In this sense one speaks of the diameter rather than a diameter (which refers to the line itself), because all diameters of a circle or sphere have the same length, this being twice the radius r.
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{displaystyle d=2rquad Rightarrow quad r={frac {d}{2}}.}
For a convex shape in the plane, the diameter is defined to be the largest distance that can be formed between two opposite parallel lines tangent to its boundary, and the width is often defined to be the smallest such distance. Both quantities can be calculated efficiently using rotating calipers. For a curve of constant width such as the Reuleaux triangle, the width and diameter are the same because all such pairs of parallel tangent lines have the same distance.
For an ellipse, the standard terminology is different. A diameter of an ellipse is any chord passing through the center of the ellipse. For example, conjugate diameters have the property that a tangent line to the ellipse at the endpoint of one of them is parallel to the other one. The longest diameter is called the major axis.
The word “diameter” is derived from Greek διάμετρος (diametros), “diameter of a circle”, from διά (dia), “across, through” and μέτρον (metron), “measure”. It is often abbreviated DIA, dia, d, or ⌀.

Width
In geometric measurements, length is the most extended dimension of an object. In the International System of Quantities, length is any quantity with dimension distance. In other contexts, length is a measured dimension of an object. For example, it is possible to cut a length of wire shorter than the wire’s width.
Length may be distinguished from height, which is vertical extent, and width or breadth, which are the distance from side to side, measuring across the object at right angles to the length. Length is a measure of one dimension, whereas area is a measure of two dimensions (length squared) and volume is a measure of three dimensions (length cubed). In most systems of measurement, the unit of length is a base unit, from which other units are derived.

Diameter (noun)
Any straight line between two points on the circumference of a circle that passes through the centre/center of the circle.

Diameter (noun)
The length of such a line.

Diameter (noun)
The maximum distance between any two points in a metric space

Diameter (noun)
The maximum eccentricity over all vertices in a graph.

Width (noun)
The state of being wide.

Width (noun)
The measurement of the extent of something from side to side.

Width (noun)
A piece of material measured along its smaller dimension, especially fabric.

Width (noun)
The horizontal distance between a batsman and the ball as it passes him.

Width (noun)
The use of all the width of the pitch, from one side to the other.
“Manchester United like to play with width.”

Width (noun)
the measurement or extent of something from side to side; the lesser of two or the least of three dimensions of a body
“the yard was about seven feet in width”

Width (noun)
a piece of something at its full extent from side to side
“a single width of hardboard”

Width (noun)
the sideways extent of a swimming pool as a measure of the distance swum.

Width (noun)
wide range or extent
“the width of experience required for these positions”