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Yes, the assisting crane is a mobile crane where the boom pivots up and down about the base. Some have booms that need to be disassembled piece-by-piece, so you just pivot it down until the top is resting on a flatbed trailer, then unbolt it.

These mobile cranes can reach just as high as tower cranes, but can't carry as much, can't reach out over the tops of buildings, and take up more room on the ground. Hence the need for tower cranes.

Another interesting advatange of tower cranes is that they can grow/shrink themselves by using a special jig; that's how they're able to negotiate 200 story skyscrapers. That jigs tends to be too convulted to use for smaller projects, in which case they use a mobile crane as seen here.

200 story skyscrapers? In some other universe?

https://www.youtube.com/watch?v=rLcxROazqy8

This is Liebherr LG 1750 which is a "mobile crawler crane" that is completely self-propelled and self-assembling.

This was my same thought. If it takes a crane to build another crane, then why not use the first crane to begin with?

Not all cranes are the same. Some cranes can only assemble other cranes. Some cranes can't be built without another crane's help.