Residential duplex outlets come in a variety of styles.
The two most common seem to be:
Does it matter? Is the difference purely aesthetics, or is there any technical difference?
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The outlet on the left will accept both 15 amp and 20 and compatible
The outlet on the right is a 15 amp only outlet
The left is formally a: NEMA R5-20R
The right is formally a: NEMA R5-15R
The designation is for the pin pattern alone
The shape of the 'Square' vs. the 'Double D' outline is cosmetic only
The square shape is typically called 'Decora' - the word Decora is a trademark of Leviton. But it is used generically.
At this point, I think you can actually get more things in the Decora pattern footprint. GFI, Motion sensors, and other devices come in the Docra pattern, but are not seen often, if at all in the double d format.
Both of the devices you show are in a 'Single Gang' and 'Duplex' format.
'Gang' is a trade unit of measure (2.812" x 1.75") and 'Duplex' refers to the fact that there are two of them.
The 'NEMA' (knee-ma) is the acronym for National Electrical Manufactures Association. Lots of information, but being a standard, it lacks the symmetrical debates that occur with the NEC
There are two main differences between the outlets shown.
First is the left one is a 20 amp outlet and accepts both 15 and 20 amp plugs. You can tell because the left prong is T shaped. The right outlet is 15 amp only because the left prong is just vertical.
Second is the visual design: two cropped circles on the left vs. a single rectangle (a.k.a. decora). This is purely aesthetic and has no functional difference.
Since you are more concerned about the plates, There are a number of standard plate designs designed to accommodate different devices in the box.
Decora which you show on the right works with not only regular outlets which you show, but with gfic outlets, switches, dimmers, fan controls, timers, motion sensors and more.
Duplex which you show on the left is the most common for outlets, but has also been used for mixed switch (one switch, one outlet), double switch, switch with indicator, telephone, tv antenna (I have seen both 300 ohm and 75 ohm) and keystone. All of these special applications are also available in decora except 300 ohm tv.
Rounding out the most common wallplates is the switch plate, which accommodates a single switch in a small rectangular cutout. Also available in this form factor are dimmers and indicator lights.
The less common wallplates with a few exceptions are single round cutouts of various sizes to accommodate single outlets of various descriptions.
The exceptions are the old style double button switch which has two round cutouts, keystone plates used for communications, plates with integral specialty jacks and oddball plates that are matched to the equipment.
When choosing, cost and aesthetics are your typical consideration (assuming the plates are purchased at the same time as the outlet they are used with when you put one device in a box (single gang). When you put two devices in a box (double gang) you are more limited, although the common pairings are available. With three devices (triple gang) and wider you are limited to switch and decora style plates.