If you know C, you will love Arduino. It's very easy to "to speak to a web API". They are usually low power 8-bit CPUs with 16 MHz, exactly right for small projects like blinking some lights that should run on battery for years.
JavaScript/Python/etc on RaspherryPi or this new Mozilla thingy are interesting too, though will drain your battery faster. You need an OS on a 32bit CPU. If it's connected to the web you should also update the software stack.
Isn't it the opposite? If you aren't comfortable with C, Arduino has a simple enough API and good enough tutorials+documentation that you can still make it work.
If you are comfortable with C, just about everything is better than arduino because you'll get debugger support and full control over all the goodies like clocks, charge pumps, DMA controllers, asynchronous ADC operation, pin multiplexers, USB/Ethernet controllers, etc.
I live in the middle. For quickie little projects like my NMEA protocol converter, the Arduino tools & IDE are fine, if barely adequate as the project has grown.
However, I have another project where Atmel Studio is a better choice. So I import all the Arduino libraries I need (there's a plugin that can do this automatically, but it's buggy), and then just build it as a regular C++ project but I get "free" use of the libraries without the crappy Arduino IDE.
The nice thing about Arduino is it's so popular that the economies of scale let me buy hardware cheaper than I could build it myself and pretty much all the libraries I need are already written. Although I have had to debug a few of them!! The bad thing is their terrible interface that's really made for tiny programs only.
It depends what you want/need. Initially I thought like you too, I coded embedded C stuff years ago. Though for quick prototyping and hobby things, it's a lot faster to write C code with macros (=Arduino), you can put together a nice prototype in hours. If you want to sell your product, you can always rewrite it later.
I thought Arduinos didn't come with Ethernet on the chip, and instead you have a Heath-Robinson addon board which you speak SPI to another microprocessor which has the TCP/IP stack, MAC and PHY on?
The only thing I've seen that seems to really be viable for remote battery operation is the ESP2866. I have a Pi in my cupboard reading the electricity meter.
I really ought to learn Arduino, simply because it's so popular.
There several Arduino models that have an ethernet port on board and there is the official Ethernet Shield for other models like UNO, Mega. The Arduino standard library supports the common solutions out of the box. So no, you don't have to bother with low level stuff (but optionslly, you can write very low level C and shift registers and speak SPI directly, if you want).
JavaScript/Python/etc on RaspherryPi or this new Mozilla thingy are interesting too, though will drain your battery faster. You need an OS on a 32bit CPU. If it's connected to the web you should also update the software stack.