Difference between grease
Each kind of grease will perform best at a certain temperature. And is designed for a certain speed of movement of the parts it is used at to reach and not exceed that temperature.

So yes, it is worth it to use the special grease for bikes or get an other kind of grease that is designed for the speeds/temperature your parts will get.

I found that when a friend opened the ball bearings of one of the rear wheels of my recumbent trike and showed me dried out grease, which had just never gotten warm enough to spread through the bearing. He replaced it with grease for slow moving ball bearings. From then on the trike did run much better.

For the grease that does not move, there are also different ones. And again designed for different work. For those parts of your bike where you hardly ever want movement, you want grease that does not stick when it gets old. It should not go soapy or sticky because it changed character.