What is the difference between city sidewalk concrete and backyard concrete?

The sidewalk concrete is indestructible.

The junk used for backyards and home walkways if junk that cracks and splays after a year or two. Why?

A lot will have to do with the prep work of the ground before you pour. Also, concrete has various tensile strength. The mix of rock and sand, the amount of water. All makes a difference. Also, what reinforcement (often re-bar) is used.

Following that, is trees near, that will cause root problems.

There is no special certain City Sidewalk concrete or Backyard concrete.

Concrete comes in various strengths... based on application. Jim-Bob might mix the cheapest stuff he can find in a wheel barrow with too much water slop it in between 2X4's way too thin and without any reinforcement, without properly screeding or floating the concrete, etc, City Sidewalks are usually built by people that actually know what they are doing.

Learning how to work with concrete correctly takes a lot more time than 5 minutes reading the lable on a sack. IT takes years to be really good at it, if you are taught by people that know what they are doing as well.

If you are talking about concrete on private property versus concrete in public rights of way, the difference can be night and day. The backyard concrete may not be any particular design, placed with a high slump (too much water), placed on unstable soil/foundation, not thick enough, allowed to dry out during cure period, etc. Public concrete was probably placed by a contractor or public works crew subject to specifications and inspection. One of the biggest differences in durability is probably the air entrainment usually added to public concrete, which vastly increases the performance in areas subject to freeze/thaw cycles.
Concrete has high compressive strength and much lower low tensile strength, in comparison, and reinforcing steel is placed in areas where higher tensile stresses are encountered.
The sidewalk concrete is not indestructible, basically nothing is indestructible. With the correct mix, and the correct additives and design, and good specifications, foundation, and curing, backyard concrete can be superior to sidewalk concrete.