Looking to expand MacBook Memory: Should I buy a 128/256GB SD Card or Flash Drive?

That does not provide you with more memory. It provides you with more disk storage space. Memory (RAM) and storage space are not the same types of "memory."

Flash drives and SD cards are just different forms of storage. SD cards are smaller than USB flash drives while flash drives are usually faster and less expensive per GB.

Note that you cannot install Windows on either nor can you boot Windows from either.

Yes, they can be formatted and partitioned. If you use them for both Windows and the Mac, then I would suggest just using two of them. One prepped for the Mac and the other for Windows.

Use Disk Utility to partition and format for the Mac. You can use Disk Utility to format a Windows volume using FAT or ExFAT. If you want to use NTFS then you need to do the formatting on a Windows machine.

They can be treated just like any other drive, and they work the same.

If you own the 13" MBA it has an SD card slot built in. Transcend makes a unique SD RAM card that fits flush in the slot (other cards stick out about a 1/4 of an inch which isn't safe). The largest capacity right now is 128GB. The advantage of this over an external SSD drive is that you don't have anything extra to carry about - the card can be left plugged in all the time. The main disadvantage is that the SSD drive is much faster. However you'll probably be using this extra storage for music and other data so the speed shouldn't be a consideration. I recently purchased one of these for my 2011 MBA and am quite happy. And yes, you can partition it.

If you buy a normal SD card it will stick out. If you are lucky nothing bad will happen. If you are a little unlucky something will bump the SD card and damage it - maybe enough so it can't be read any longer. If you are very unlucky it will damage the logic board of the computer. If you are truly unlucky the bump knocks the computer to the ground and not only the logic board is damaged but also the display. I've seen the aftermath of each of those circumstances at least once. I was also pleasantly surprised that the Transcend card was faster than the card I'd been using.

I just unpacked mine and put it in, then formatted it without any problems. Unfortunately I’m at work and the computer is on its way to Apple for a new battery so I can’t look at my SD card. Many of them have a little tab on the side to lock it so the disk cannot be accidentally overwritten. Check to see if that’s your problem.