We can say arduino is an other flexible controller option like a ROS, LinuxCNC, raspberry and so on.
Arduino hardware components are cheaper in relation with other controller architecture and programming language is easy. Arduino has greater academic applications.
One disadvantage of the Arduino Microcontroller is that its resolution is only 10 bits. But you can expect very high resolution for a less that 10 USD Microcontroller. Furthermore, you can add modules to increase the resolution to up to 16 bits for 15 USD.
The main advantage is its cost and its modular capabilities.
There are a lot more microcontrollers than the few operating on some "flavor" of the "Arduino board family", but the Arduino board family ...
has a common IDE which is really easy to use
has a lot of peripheral support - including 3rd parties
has a lot of sample applications to start with
does easily scale between different members of the family - including changing the microcontroller family the individual boards are based upon.
Contrasting to that, more "general microcontrollers" usually come with their own "ecosystem", often including some IDE. But there hasn't been much effort to make projects "portable" and capabilities of the individual approaches may differ.
As an overall assessment, while the Arduino ecosystem may or may not give the maximum from the microcontroller(s) used, it offers high portability and programming convencience - at the cost of the last of performance. In contrast, programming "general microcontrollers" ("bare metal") requires a lot more effort, offers little portability and less "comfort"; doing so unleashes the full performance of the individual microcontroller (no cosiderations about portability, a unique configuration scheme etc.).
Low cost boards and peripherals (when sourced from China :)
It's simple (i.e. add abstraction to underlying MCU's "nuts & bolts" many times for the sake of efficiency/processing speed), but not too simple: so-called sketches are actually C/C++ code and you need some real programming skills for some real job
Disadvantages:
It's mostly still AVR (8-bit) "eco-system" (and +5 V). There is many claims that other (e.g. ARM) architecture are supported but you'll find pretty soon that even 32-bit boards designed by Arduino team (e.g. Due, Zero, MKR) are not supported in a similar way to 8-bit (Uno, Leonardo, Mega2560): they are still second choice
If you need more processing power and working memory be ready to kiss goodbye Arduino environment pretty soon
Arduino is a microcontroller kit for beginners which facilities the hardware and software interaction with a micronctroller chip in a sort of plug and play. Arduino is a kit whereas microntroller is a chip; certainly, you can't compare a kit to a chip As an example Arduino Uno is based on ATmega328P chip which is manufactured by Microchip Technology co.
But we can compare between Ardunio and another starter kit as raspberry pi or Beaglebone of Texas Instrument as we can compare a microelectronic chip to another according to the datasheet.
Firstly, Arduino is the name of the board powered with AtMega168 or AtMega32 microcontroller.
The main advantage of Arduino is its simple programming language. It's designed to make new user to get a glimpse and boost their confidence of programming. Originally, Arduino was designed for school kids to get a taste of programming and hardware design. It's suitable for making a demo model of any idea.
However, for more advanced and real time projects, Raspberry pi, Beaglebone, and other advanced boards are the only option left with as Arduino is not designed to handle large complexity that comes with advanced projects.
So, in conclusion, for design of basic embedded system design (Software + Hardware) , and who are just beginners, Arduino is the best compared all other boards available in the market.
However, after you get grip and command on embedded system design and are in the mode of designing advanced systems, Arduino is not suitable compared to other available options.
I agree with Rahuls answer, the Arduino is great for making a first break into microcontrollers, but if you are really looking to build your own instrumentation then I would go straight into the AVR-8 microcontrollers used on the same boards. With Chinese PCB production costs so low at the moment, its also very cost effective, and save messing around with jumper cables etc.
I would like to add to the colleagues that Arduino is a micro controller development board based on the microcontroller ATmega. The microcontroller development board has input and output ports both analog and digital. To complete the discussion i added a link where the Arduino board is described in detail which may be useful: https://www.elexp.com/PDFs/01ARDUNO.pdf
The paper also introduces Schematic & Reference Design.
Arduino has some sound advantages, such as large user community, free and broad ranges of libraries of codes, relatively low cost components, and so forth.
Its disadvantages are its small and a user has to work in a relatively small (or rather tiny) space. In many broad and multi-purpose projects, its required to look for third party sources in addition to Arduino scripts. Another disadvantage is that in many cases, some errors remain quite persistent and error messages may not be quite useful.
Its usefulness for learning purposes and developing really cool boards is very sound.