user81075: I'm trying to understand when one would use an Arduino module rather than directly a sensor.
Here's an example to contextualize my question: I need to use a Hall sensor to measure the number of rotations per minute of a magnet mounted on a computer fan. I see that I can either buy the Hall sensor and wire it to the Arduino (Hall-Effect Sensor Analog 49E), or alternatively I could buy a sensor module (e.g., Digital Hall Effect Sensor Module or Linear Hall Effect Sensor Module). What is the advantage of buying modules, rather than the sensors directly?

user68595: Since nobody else has addressed your example directly, and to use it as an example, in addition to the other observations about these two items, let's also take note of :
Hall-Effect Sensor Analog 49E

The 49E sensor can detect the magnetic N/S pole and relative strength of a magnetic field and outputs an analog voltage


Vcc Range    2.3 – 10V
Output : Analog 1.9mV/GS (± 0.2mV/GS)

Digital Hall Effect Sensor Module

3144 sensor detects presence of magnetic field and outputs logic level.


Vcc Range    4.5 – 24V
Output :  Open Collector

Linear Hall Effect Sensor Module

Detects N/S poles and relative strength of a magnetic field.


Vcc Range 3.3 – 5V
Output : Analog 2.5mV / G PLUS digital output
Includes 49E sensor plus everything in the schematic below :


So the sensor you've linked provides an analog signal, allowing you measure the magnetic field strength.  The digital module provides only an open-collector output (ie: ON/OFF) which indicates either the absence or presence of a magnetic field above some threshold of intensity. These are entirely different functions which you would use for entirely different applications.  The digital module also works with VCC up to 24V, which makes it handy to integrate with 24V automation systems.  The linear module provides an analog output as well as some indicators and an additional digital output, but is capped at 5V Vcc (whereas the raw module accepts up to 10V).
Why would you choose either?  What do you need to do?  Do you want to check for the presence of a magnetic field only, or do you also want to measure its intensity?  Or both?  The modules do some of the work for you and, in the case of the linear module, actually give you an additional output that the raw sensor does not.  It's up to you to decide which of these serves your needs.

user37604: It's a question of convenience and application use-case specific. Sometimes a module will give you additional functionality and/or structural stability, just by providing a mini-PCB (maybe with screw holes) for a low price overhead.
For example, compared to the bare element the module you posted has additional components (LED or A0,D0 on the pricier one) is easily mountable and provides standarized connection pins.
If you already have a PCB (or other circuitry) the module could be superfluous, maybe you just need the sensor itself, but often those modules come as neat packages, sometimes with controller chips speaking I²C and comparable protocols (like the DHT11/DHT22 and others).

user70044: The modules are typically easier to work with, particularly for DIY projects. They have standardized connectors, have their connections labeled and can be mounted into wiring boards. The particular sensor you have there does have quite big legs, so this may work fine without the board, but this is often not true. Consider sensors such as the BME/BMP types (BME680 data sheet). They're far to small to be used directly without a PCB. And they require some external components to work correctly, which are already included on the module. So for experimental purposes, the module is the way to go.

81075: <p>I'm trying to understand when one would use an Arduino module rather than directly a sensor.</p>
<p>Here's an example to contextualize my question: I need to use a Hall sensor to measure the number of rotations per minute of a magnet mounted on a computer fan. I see that I can either buy the Hall sensor and wire it to the Arduino (<em><a href="https://protosupplies.com/product/analog-hall-effect-sensor-49e/" rel="nofollow noreferrer">Hall-Effect Sensor Analog 49E</a></em>), or alternatively I could buy a sensor module (e.g., <em><a href="https://protosupplies.com/product/digital-hall-effect-sensor-module/" rel="nofollow noreferrer">Digital Hall Effect Sensor Module</a></em> or <em><a href="https://protosupplies.com/product/linear-hall-effect-sensor-module/" rel="nofollow noreferrer">Linear Hall Effect Sensor Module</a></em>). What is the advantage of buying modules, rather than the sensors directly?</p>
 68595: <p>Since nobody else has addressed your example directly, and to use it as an example, in addition to the other observations about these two items, let's also take note of :</p>
<h4>Hall-Effect Sensor Analog 49E</h4>
<blockquote>
<p>The 49E sensor can detect the magnetic N/S pole and relative strength of a magnetic field and <strong>outputs an analog voltage</strong></p>
</blockquote>
<ul>
<li>Vcc Range    2.3 – 10V</li>
<li>Output : Analog 1.9mV/GS (± 0.2mV/GS)</li>
</ul>
<h4>Digital Hall Effect Sensor Module</h4>
<blockquote>
<p>3144 sensor detects presence of magnetic field and <strong>outputs logic level</strong>.</p>
</blockquote>
<ul>
<li>Vcc Range    4.5 – 24V</li>
<li>Output :  Open Collector</li>
</ul>
<h4>Linear Hall Effect Sensor Module</h4>
<blockquote>
<p>Detects N/S poles and relative strength of a magnetic field.</p>
</blockquote>
<ul>
<li>Vcc Range 3.3 – 5V</li>
<li>Output : Analog 2.5mV / G PLUS digital output</li>
<li>Includes 49E sensor plus everything in the schematic below :</li>
<li><a href="https://i.stack.imgur.com/ZU3Ls.jpg" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/ZU3Ls.jpg" alt="Linear Hall Sensor Module Schematic" /></a></li>
</ul>
<p>So the sensor you've linked provides an analog signal, allowing you measure the magnetic field strength.  The digital module provides only an open-collector output (ie: ON/OFF) which indicates either the absence or presence of a magnetic field above some threshold of intensity. These are entirely different functions which you would use for entirely different applications.  The digital module also works with VCC up to 24V, which makes it handy to integrate with 24V automation systems.  The linear module provides an analog output as well as some indicators and an additional digital output, but is capped at 5V Vcc (whereas the raw module accepts up to 10V).</p>
<p>Why would you choose either?  What do you need to do?  Do you want to check for the presence of a magnetic field only, or do you also want to measure its intensity?  Or both?  The modules do some of the work for you and, in the case of the linear module, actually give you an additional output that the raw sensor does not.  It's up to you to decide which of these serves your needs.</p>
 37604: <p>It's a question of convenience and application use-case specific. Sometimes a module will give you additional functionality and/or structural stability, just by providing a mini-PCB (maybe with screw holes) for a low price overhead.</p>
<p>For example, compared to the bare element the module you posted has additional components (LED or <code>A0</code>,<code>D0</code> on the pricier one) is easily mountable and provides standarized connection pins.</p>
<p>If you already have a PCB (or other circuitry) the module could be superfluous, maybe you just need the sensor itself, but often those modules come as neat packages, sometimes with controller chips speaking <a href="https://en.wikipedia.org/wiki/I%C2%B2C" rel="noreferrer">I²C</a> and comparable protocols (like the <a href="https://lastminuteengineers.com/dht11-dht22-arduino-tutorial/" rel="noreferrer">DHT11/DHT22</a> and <a href="https://arduino.stackexchange.com/questions/53882/communicating-with-magnetic-sensor-tlv493d-a1b6-over-i2c">others</a>).</p>
 70044: <p>The modules are typically easier to work with, particularly for DIY projects. They have standardized connectors, have their connections labeled and can be mounted into wiring boards. The particular sensor you have there does have quite big legs, so this may work fine without the board, but this is often not true. Consider sensors such as the BME/BMP types (<a href="https://cdn-reichelt.de/documents/datenblatt/X200/BME680_DB-EN.pdf" rel="noreferrer">BME680 data sheet</a>). They're far to small to be used directly without a PCB. And they require some external components to work correctly, which are already included on the module. So for experimental purposes, the module is the way to go.</p>
