For PC gaming on external storage, should I use Flash or HDD (hard disk drive)?
Solution 1:
I believe neither are good options.
Running Fallout 4 will most likely keep the drive active for the entire duration of your playtime. Neither external flash nor HDD have sufficient heat dissipation to handle that. You will most likely destroy your drive very quickly.
Unfortunately you probably won't know this is happening because the external drive are typically housed in a box. You will have to take the device apart to notice this.
If you are still considering this, look for a high quality, metal external drive, with an active heat sink (has a fan) and also heat sink fins on the box. However, these are significantly more expensive.
As an alternative, the 1.8"/2.5" format is usually safer. The HDD itself are designed for laptops, and usually generates less heat because the "boxed in" environment are exactly how they operate in a laptop. However, do look for a metal one that feels cool to the touch (heat dissipation), and avoid plastic ones.
Note: your HDD itself will most like be fine, but the enclosure is usually a cheap plastic box with no holes. Cheap plastic electronic + heat is usually a combination to stay away from :)
Solution 2:
These are your options in order from best to worst:
Solid State Drive (SSD)
A solid state drive will provide much faster access than a mechanical hard drive. Amazon shows that in August 2015 you can get 250GB external SSD for around $150. According to Game Debate Fallout 4 requires 28GB of hard drive space so you could target an even smaller drive to reduce costs. You may also choose to assemble your own drive by purchasing an internal SSD and a 2.5 inch USB 3 Enclosure to contain the drive. The enclosure should be able to be found for around $10.
You should be able to assemble a 120GB drive for under $70 and a 60GB drive for around $50 from either Amazon or your local computer parts store. My examples above are assuming you are in the USA, but as another example here in Australia I could assemble a 120GB SSD in a USB 3 enclosure for under $90 AUD.
Powered 3.5 Inch Mechanical Drive
A powered external drive will be larger than a portable drive (and not very useful if you are on the go) but it will provide ample space (multiple terabytes) and should perform similar to a mechanical hard drive that you would have found inside a desktop PC a few years ago. Note that these days most "gaming" PCs will be built with Solid State Drives internally.
"Pocket" 2.5 Inch Mechanical Drive
A pocket drive is still usable, and can be found quite readily and cheaply. Be prepared for long loading times and transitions, and also long install times when using Steam as it will have to pause during downloading to wait for all the files to be unpacked to the disk. I had significant success using one of these drives a few years ago to play games under Boot Camp on a MacBook Pro, allowing me to install lots of games but keep a small (20GB) Boot Camp partition.
Flash Drives / Memory Cards
Even though flash drives have no moving parts like an SSD, there is still a large performance difference between the two types of drives, especially when reading or writing multiple files at the same time. The performance can be improved by purchasing a high quality (and expensive) memory stick, but at that stage you would be far better off getting a 120 or 240GB SSD.
Solution 3:
Realistically, the speed of the drive won't affect gameplay much. The main benefit to a HDD would be capacity and price, as an external SSD will not run nearly at the speed the price would imply when connected via USB 3.0. I gamed for quite a while on a USB 2 external HDD, and I never experienced any major issues, save for slightly extended load times.