A: Let's start with defining what these devices are.
HARD DISK DRIVE (HDD)
An HDD is an internal or external drive that uses rotating disks that are covered with a metal oxide (same as on video tape or cassette tape). This technology has been around since before the dawn of the PC. The cost is relatively low - an external portable USB HDD that stores 5TB of data can be purchased for under $150. They are relatively slow - around 1/4-1/10 the speed of a Solid State Drive.
SOLID STATE DRIVE (SSD)
An SSD is an internal or external storage device that uses electronics (transistors and capacitors) to store data. This is newer technology. The cost is relatively expensive - an external portable USB SSD that stores 1TB of data can be purchased for under $250. They are relatively fast - from 4-10x faster than an HDD.
WHICH TO USE FOR BACKUP?
There is no right or wrong answer to the question. The trade-off usually comes down to either saving money up front by purchasing the less expensive HDD, versus saving money down the road at the time of data recovery by purchasing the much faster SSD. Given that an SSD is 4-10x faster than an HDD, when recovering data (or rebuilding the entire drive), it will take 1/4 as long to do so when using an SSD.
Generally, my recommendation is to go with an SSD. Although it may cost $100-$400 more at time of purchase, it may take only 2 hours to fully rebuild a corrupted hard drive, as opposed to 8 hours using an HDD.