I'm 5 foot 1, and I'm stuck between a two bikes one 29 inch wheel and the other 26 inch hybrid bikes (more towards road bike style) for errands around a metropolitan city. Both bikes have a small frame but I'm unsure what the potential advantages and disadvantages are between these two bike tyres in my context.
Honestly wheel size doesn't really matter for your use.
I'm over a foot taller than you, my current main hybrid/commuter/errand bike is 29" (700C) as are my mountain bike and tourer. But I've had a few 26" urban errand bikes and they did the job perfectly well too.
What's far more important is that you're comfortable on it, including stopping and starting. Both could fit perfectly but still ride differently. As an example, some people prefer a saddle set further behind the pedals if they're stopping a lot, to make it easier to put their feet down while seated. Some of us find that riding position unnatural. So try both and see if there's a significant difference in comfort. Even just inside the shop or its car park can be instructive.
At 155cm you’ll probably need the smallest frame size for a bike with 622mm (28"/29") wheels. The smallest frame size often has to make some compromises to allow 622mm wheels to fit.
You’d have to look at the geometry specification of both bikes and compare them to find out. Of course a test ride would also be great. If the bike with 622mm wheels is not too big for you, you don’t hit the front wheel in tight turns and it handles nicely then it’s totally fine to get it (assuming wheel size is the only difference and all the components, weight etc. are comparable).
A bigger wheel will roll slightly better over obstacles or rough ground. However the effective wheel diameter also depends on tyre width. A 559mm (26") wheel with wide MTB tyres will actually have a bigger effective diameter than a 622mm wheel with narrow road bike tyres. Wider tyres can also be run at lower pressures which allows them to roll over obstacles more easily (and comfortably). Even with same tyre width the difference is small (it’s only a ~10% difference in diameter after all). MTBs used to have 559mm wheels for decades and only switched to 622mm recently after years of debate and back-and-forth. Other types of bike are still available with both wheel sizes.
I can tell you my story as I'm your height. I have a bike that has 24-inch wheels, the smaller bike fits me better in many situations. I think that size of the bike should not be a prerequisite.
I recommend you to ride few bikes from all sizes and types, then you will have a taste of what you will need.
I have assembled three hybrid bikes based on MTB frames with slick tires and used them for commuting, parcel delivery and multi-day road travel. Two of them where 26" and one of them is 29" (700c x 42).
I would argue that all these bikes where virtually the same, with very little differences among them. None of them made any difference towards riding or ease of use. The one with the 700c wheels should theoretically be a little bit more comfortable regarding small bumps in the road and over cracks in the pavement, but I doubt that is really perceived or whether it is placebo. Definitely it's more influenced by the specific tire pressure used.
Since they where all built for myself, I used the same saddle-handlebar distances. In all of these bikes I used the same model of aluminium rigid fork.
I only experienced a bit of an issue with pedal strike when leaning too much in one of the 26" bikes, but some of that was due to the fork being too short for the geometry of the frame, which placed the bottom bracket too low. (Bottom bracket height does not depend exclusively on wheel size, chainstays are sloped differently in different frames)
The other subtle difference is that the 700c bike has greater overall length, despite having almost exactly the same axle to axle dimension. This does not influence riding too much, but in storage and transporting of the bike. For example, the 26" one fits with room to spare in my pickup truck's bed, while the 700c one has to be fiddled with a bit in order to fit.
One difference is the pedal clearance that matters on
- paths with very low raised borders (common between path and grass in some places),
- slow, very tight turns with readiness to stop (the "wrong foot" may soon be needed on the ground, so keeping it down is not that irrational).
- bumps and water hose protectors across the road (you need to remember keeping both pedals level when crossing these)
When I changed from 29" to 27.5" bicycle, I managed to hit all three of these multiple times before I learned to pay special attention how to avoid them, but no earlier than damaging a pedal twice, once beyond repair. Hence I would say no, not just a placebo.
From the other side, with smaller wheels you are closer to the ground that makes a truly beginner to feel more comfortable psychologically. Unprepared (unexpected) stops are easier and once you are at still stand you will never fall that is not true for a beginner on large wheels. But here, also, even basic riding experience can compensate.
Bigger wheels also have less rolling resistance on rough, uneven surfaces like bricks (in old town) or gravel. If you are really sure that you will never ride from tarmac, this is less important, but anyway disadvantage. Even a center of a big city may have a gravel path for bicycles.
As a general rule, unless you have some compelling reason to want smaller wheels, you usually want the largest wheels you can get while still having a well designed frame that properly fits you.
There’s not one single reason for this, but a combination of benefits. The two major advantages are higher pedal clearance (higher rear hub means higher BB, which means higher pedals unless you get longer cranks, which are uncommon) and a better angle of attack when riding over things (which in turn means a smoother ride). Larger wheels also have slightly higher traction on solid flat surfaces (at the same tire pressure, a larger wheel will have a larger contact area with the road), which is generally a good thing for your described use case.
The caveat here is that your height means you will have trouble finding a good frame that fits you properly and actually fits 29-inch wheels. In this case, I would actually recommend looking for a bike with 27.5-inch wheels (ISO 630mm or French 650B). 27.5 inch wheels are a common size for mountain bikes, so it’s generally not too unusual to find them on hybrid bikes if you actively look for them. Regardless of what wheel size you go with though, make sure the bike itself fits you properly, as that’s important not only for reasons of comfort, but also for efficiency.
This is tangential to your question, but given you mention primarily urban utility usage, I would strongly recommend investing in a bike that has disc brakes instead of rim brakes (I noticed both bikes in your pictures have rim brakes, at least on the rear wheel). They’re a bit more expensive to maintain, but generally offer much more consistent performance independent of the weather conditions (rim brakes tend to perform noticeably worse in wet weather).