Difference between projection and datum?

Geographic coordinate systems (lat/long) are based on a spheroidal (either truly spherical or ellipsoidal) surface that approximates the surface of the earth. A datum typically defines the surface (ex radius for a sphere, major axis and minor axis or inverse flattening for an ellipsoid) and the position of the surface relative to the center of the earth. An example of a datum is NAD 1927, described below

Ellipsoid        Semimajor axis†          Semiminor axis†   Inverse flattening††
Clarke 1866     6378206.4 m              6356583.8 m             294.978698214

All coordinates are referenced to a datum (even if it is unknown). If you see data in a geographic coordinate system, such as GCS_North_American_1927, it is unprojected and is in Lat/Long, and in this case, referenced to the NAD 1927 datum.

A Projection is a series of transformations which convert the location of points on a curved surface(the reference surface or datum) to locations on flat plane (ie transforms coordinates from one coordinate reference system to another).

The datum is a integral part of the projection, as projected coordinated systems are based on geographic coordinates, which are in turn referenced to a datum. It is possible, and even common for datasets to be in the same projection, but be referenced to different datums, and therefore have different coordinate values. For example, the State Plane coordinate systems can be referenced to either NAD83 and NAD27 datums. The transformations from geographic to projected coordinates are the same, but as the geographic coordinates are different depending on the datum, the resulting projected coordinates will also be different.

Also, projecting data may result in a datum conversion as well, for example, projecting NAD_1927 data to Web Mercator will require a datum shift to WGS 84. Similarly, it is possible to convert data from one datum to another without projecting it, as with the NGS's NADCON utility, which can shift coordinates from NAD27 to NAD83.

Example of a point's coordinates referenced to different datums

Coordinates referenced to NAD_1927_CGQ77

19.048667  26.666038 Decimal Degrees
Spheroid: Clarke_1866
Semimajor Axis: 6378206.4000000004
Semiminor Axis: 6356583.7999989809

Same point referenced to NAD_1983_CSRS

19.048248  26.666876 Decimal Degrees
Spheroid: GRS_1980
Semimajor Axis: 6378137.0000000000
Semiminor Axis: 6356752.3141403561

You will obviously get better answers from textbooks, but here is an simple explanation:

Map Projection: It is a method for representing a spherical or curved surface on a flat plane.

Datum: It is the reference or origin based on which measurements are made.

After struggling with this question ten years ago, and finding many confusing things written about the topic, I published a brief article in Directions Magazine that presented an answer as simply, plainly, and accurately as I could make it. The following is excerpted from that article.

Reprojecting geographic features

Two things must happen when you draw a map: features in the real world must be "georeferenced" to a spheroid and the spheroid must be projected onto the paper.

The spheroid models the shape of the earth's surface. It is an idealization that does not account for local changes in topography.

Georeferencing assigns locations (in three dimensions!) to points on a spheroid.

Projecting is an operation that mathematically distorts and shrinks a portion of the spheroid onto flat paper. Projecting can be undone ("inverted"). "Unprojection" expands a feature on a map and plasters it back onto the spheroid.It, too, is a mathematical operation.

Georeferencing is done with a datum. A datum is usually given by a starting point and direction: it specifies where a clearly identifiable point on earth (the base point) should appear on the spheroid and it shows where a base direction, such as north, points on the spheroid at the base point. The base point and direction allow surveyors to determine the distance and angle of any other point on the earth. Moving in the corresponding direction on the spheroid for the same distance determines where the new point should go on the spheroid.

Spheroids have coordinates. They are latitude and longitude. (Geodetic) latitude is the angle made by a vertical line to the horizontal. It is not necessarily the same angle made by "straight up," because the latter is distorted by gravitational variation over the earth. It is not necessarily the angle made by a line to the center of the earth, because most spheroids have an elliptical cross-section, not a circular one.

Therefore, georeferencing endows points near the earth with latitude, longitude, and height coordinates.

(Subsequent sections discuss Change of datum, How to relate two maps, The wrong way to do it, and North America is a special case.)