I am looking to buy another lens for my Sony a6000 for my wife for Valentine's Day. We have the kit lens, a zoom lens, and the SEL35F18. I am looking at either the Sony 20mm F/2.8 Pancake (SEL20F28) or the Rokinon(Samyang) 12mm F/2.0 Ultra Wide Angle Lens (RK12M-E-SIL). I know the second lens is manual focus but unless the item is very close it will not require adjustment. The goal is take vacation photos that will include both people and scenery. What are some of the pros/cons of each and which would be more suitable to my need? I know they are both wide angle but is going to 12mm that big of a difference.
The only real advantage of the SEL20F28 over the SELP1650 for you is probably going to be the reduction in size and weight. This can be an important consideration when travelling, especially with a camera such as the a6000 which is designed around the idea of being light and compact. Your stated use case for people/scenery photos sounds like you will mainly use it in daylight outdoors.
Your existing SELP1650 is less than one stop narrower than the SEL20F28 at 20mm. In bright light that difference is marginal since most of your shots in daylight will use narrower aperture settings such as f/5.6, f/8 or f/11. At those aperture settings any difference in lens sharpness will also be minimized between your current kit lens and the 20mm f/2.8. Typically, the main advantage of fast prime lenses over their zoom lens counterparts at the same focal length is the wider maximum aperture for use in lower light and better image sharpness when the prime is used at the same aperture as the zoom lens' maximum aperture.
The Rokinon/Samyang/Bower/Walimex/Falcon/Albinar/Opteka/Quantaray/Bell & Howell/whatever else they're calling it this week RK12M-E-SIL will give a significantly wider field of view than your kit lens at the widest 16mm setting. Don't think of it as only a 4mm difference - think of it in terms of the ratio between 12 and 16. Proportionally the difference between 12mm and 16mm is the same as the difference between 16mm and 21mm or the difference between 55mm and 75mm. Set your current zooms at 21mm and 75mm respectively. Then zoom all the way out to 16mm and 55mm respectively. How much more can you frame all the way out than with the 1.33 longer focal lengths? That's how much more you'll get with a 12mm vs. your current 16mm.
As you are already aware, the biggest disadvantage of the 12mm f/2 is that it is manual focus only. You can always use your 16-50mm kit lens at 16mm and manually focus it to see if you feel that is usable. Where a lens such as the 12mm f/2 Rokinon really shines is for landscape photography or astrophotography. Pretty much all astrophotography uses careful manual focus. A lot of landscape photographers also use very careful manual focus. So for those two uses the lack of AF is more of an advantage in terms of size/weight, complexity, and price than it is a disadvantage.
The 20mm focal lenght of hte Sony lens is covered by your kit lens. The difference is, that it is faster than the zoom lens. In other words it allows you to shoot photos of dark scenes with lower ISO or shorter times.
The 12mm focal length is not covered by your kit lens and is faster than your kit can be.
Difference between them is in different perspective, if you use same sensor coverage. The shorter focal length the bigger the effect is. See here.
If you choose the 12mm Rokinon you can shoot images you cannot shoot now and when you want to "zoom in", you can crop the image.
If you choose the 20mm Sony you can shoot one "type" of images with lower compromises.