I would summarise the distinction between registers and memories as follows:

 1. A "register" is a logic block with main function to latch and remember some data at the moment of some **clock edge** coming. Data can be flowing along many register elements, but only the selected one set got the data to remember when they are "selected", and the clock toggles. So a register usually have several more functions embedded, like "Reset" to initial state, "Set" to opposite state, and "enable/select" these functions. Without the "Select" the register cell ignores data change.

 2. A memory is usually a set of registers of different organisation in terms of parallel size (number of bits per memory word/register) and the number of addressable columns, with usually simplified access functions. For example, bulk memory doesn't have "Reset" or "Set" function, it usually has some address decoder that allows to enable one row of bits (word), which allows the data to be snatched at a clock edge, while other "memory words", being connected to the same data bus ignore the data and continue to "remember" the last write operation. This kind of memory is called "static". Minimal hardware implementation requires six transitors per one bit, the [classic 6T-architecture][1].

[![from Wikipedia][2]][2]
 
3. The main function of memory is to remember a bit of data. With substantial reduction of random-access functionality, the memory function can be accomplished by just one capacitor with low leakage, and one transistor. With some additional circuitry around and special page organisation one can have "[dynamic memory cell][3]" 

[![enter image description here][4]][4] 

Functionality of one-cap cell needs some sequence of row/column gate opening, for the charge to get into proper memory location, and a periodic "refresh" function to maintain charge levels in high or low states. It is called DRAM - dynamic random access memory, which is used in all modern computers as main system memory.

4. ROM - Read-Only-Memory, could be any of the above, with protected write function. There are re-programmable ROMs, like electrically erasable memories, or could be so-called "mask-programmed" memories where the content can't be altered.


  [1]: https://en.wikipedia.org/wiki/Static_random-access_memory
  [2]: https://i.sstatic.net/kKSzd.png
  [3]: https://webdocs.cs.ualberta.ca/~amaral/courses/329/webslides/TopicF-Memories/img40.html
  [4]: https://i.sstatic.net/FfYBs.png