Lots of answers, but most of them don't tell you much. The real story is kinda long. Get a cuppa and have a read.
First - What is the same between Windows and Linux?
Both are operating systems. An operating system provides a connection between the hardware on your computer and the software you run.
Since different computer manufacturers build their machines differently, with different hardware, if there was not a common point of reference, a software application would have to be custom tuned for each brand and model of computer, which would never work.
The OS provides an interface that can be easily adapted to different hardware while presenting the same "look" to the software, no matter who built the box.
Now what is different between the two
Windows - Windows is a commercial product developed by Microsoft. It has gone through several generations since it was introduced, and no longer resembles the product it once was.
Originally, Windows was an interface that rode on to of MS DOS, the original OS that Microsoft leased to IBM. It was not original work, but a port of CPM, Digital Researches old OS for 8 bit processors. As such, DOS was text based, cryptic in the extreme, and buggy as all get out.
Windows was Microsoft first attempt to put a graphical user interface (GUI) on top of MS DOS. It was slow clunky and prone to crashes, but it gained a foothold anyway.
In the 90's MS started a development process that resulted in Windows 95 (a better Windows) and Windows 98 (the best of breed for this line). But processors were getting faster and memory getting cheaper, and this 16 Bit OS family could not handle the larger RAM and 32 bit processors that were now common
So MS put a pretty face on Windows NT and called it XP. It took two service packs to get it right, and people complained bitterly that a lot of their old MS DOS programs did not work, and they had to spend a fortune on new software, but it became the gold standard of the PC world. 50% of the world PC still use XP
Then the processors changes again, and we moved to the 64 bit world. Windows mmorhed one more time and Vista was presented to the world. To say it was less than a hit is somewhat of an understatement, but Win 7 came out of it and is the standard today. Windows 8 is yet another change, but no one knows where it will rank just yet.
Linux had a totally different path
Unix was developed at Bell Labs, and was one of many operating systems at a time wghen every computer manufacturer was a one stop shop. You bought the hardware, OS, software programming support and hardware support all at the same place. If you were an IBM client, then if you wanted to change to DEC, you threw everything out the door and started over. All you data was transferred MANUALLY, because the machines did not communicate across families. It was Vendor Lock in with a vengeance. As Unix grew, if became one of the best "run everywhere operating systems" When the Department of Defense, sick of vendor lock in demanded that all Defense Dpt. computers run Unix, it became the standard. A lot of people worked on Unix, and it grew into the dominant OS for Big Iron - the mainframes
Linus Torvalds, a Finnish computer science student, created a port of Unix that ran on PC's. It was rough, and very bare bones, But he put it on the Internet, and soon many others were helping make it better.
Today Linux runs on everything from cell phones to supercomputers. But it never lost it's roots. It's still a Unix style OS, with all the good and bad that goes with it.
Good - Stable as a rock, efficient, versatile, open source and free to all
Bad - It is not Windows, so if you want Windows for free, you will be disappointed.
There's lots more, but I hope that helps