RS Components lists three categories of similar amplifiers: Operational Amplifiers, Differential Amplifiers, and Instrumentation Amplifiers. I think that there should be a single section called Operational Amplifiers. Why do these three separate sections exist? What is the significance of this difference?
An operational amplifier is an integrated circuit, and somewhat of a building block. The inputs (V+ and V-) of an op amp are very high impedance, and very little current goes into these inputs because of that. The general formula for the performance of an op amp is Vout=A*(V+ - V-), and A is a very large number. When wired up in feedback modes, the op amp can have many different configurations (even though the "open loop" gain is a very big number.
A differential amplifier, by definition, also functions with the relationship Vout=A*(V+ - V-), but A can be a much smaller number than in an op amp operating in open loop, and the currents into the inputs are not necessarily zero. The input impedance can be relatively low, and the input impedance to each input does not need to be the same. A differential amplifier can be built out of one or more operational amplifiers and some resistors, or it can be made out of more basic parts, like transistors.
An instrumentation amplifier is a special kind of differential amplifier. In general, it is a differential amplifier, but the input impedances on the two inputs are very high (meaning very small input currents), and the same for each input. There is usually a way to change the gain with one resistor. Very often, the instrumentation amplifier has a three op amp configuration (or the equivalent), with two op amps serving as an input stage, and the output stage is a simple one op amp difference amplifier with a reference point that can be used to move the baseline around. Because the resistors on the difference amplifier are usually laser trimmed inside the integrated circuit, common mode rejection is very high. There are also two op amp instrumentation amplifier configurations, but you lose some of the benefits (like setting the gain with one resistor).
An operational amplifier will generally be designed so that when used in a suitable feedback system the inputs will always be within some specified tolerance of being equal; most op amps are designed with high-impedance input, though how high is "high" may vary.
A differential amplifier will generally be designed to measure the difference in voltage between two inputs; differential amplifiers often have balanced but finite input resistance, and many of them can operate with input voltages significantly beyond the rails. One of the biggest problems with such amplifiers is that if the inputs are connected to things which have different resistances, the current flowing into or out of the inputs will affect them by different amounts.
An instrumentation amplifier, like other kinds of differential amplifier, is designed to measure the difference between input voltages. Unlike many other differential amplifiers, however, an instrumentation amplifier will feed both inputs directly into a high-impedance non-inverting amplifier stage with no other resistive loading. This means both inputs must be within the supply rails, but the lack of resistive loading means that no significant current needs to flow into or out of the inputs; this in turn means that even if the inputs are connected to things with different resistances, such differences will not interfere with measurement accuracy.
Operational amplifiers are general purpose devices which can be used in many applications. If the requirements are not too stringent, they can be used to make both differential and instrumentation amplifiers. Differential amplifiers are specifically designed to amplify the difference between 2 input signals. They may include specially matched resistors to help optimize this function. Instrumentation amplifiers are specifically designed for applications that require excellent DC characteristics, high input impedance, low noise and drift. They also may include on-board resistors to enable gain selection without using external components. Since differential and instrumentation amplifiers are not general purpose operational amplifiers, they are usually listed in separate categories. Although not as versatile as an operational amplifier, they provide increased performance in applications for which they were designed.
Op amps have two inputs and one output.
Instrumentation amps usually have three inputs (ref is an input) and a gain control facility, and one output.
Differential amps usually have two outputs and usually two inputs.
None are directly electrically interchangeable and this is a performance and usually functional thing.
All are aimed at solving different solutions.