I am curious what are advantages/disadvantages of using Animator instead of the Animate? Why and for what purpose the Animator has been introduced?

Who knows?

To address the question of @Kuba: There is the classical situation of making animation:

Animate[expression, {parameter, lowerLimit, higherLimit}]


There is another way:

Manipulate[expression, {parameter, lowerLimit, higherLimit, ControlType->Animator}]

• Ah, so it is Animate vs Manipulate+ControlType->Animator. – Kuba Aug 17 '18 at 9:44
• I find that Manipulate gives far greater freedom in mixing control types (sliders with buttons with pull-downs, etc.). It is extremely difficult to do that with simple Animate. – David G. Stork Aug 17 '18 at 20:21
• Animate is simply constructs a Manipulate with an Animator, which is convenient for that narrow class of applications. See the output of Animate[x, {x, 0, 1}] // InputForm. – Michael E2 Aug 18 '18 at 0:54

A great application I've used Animator for, is an animation in time while also being able to manipulate other variables simultaneously.

For instance

Manipulate[
Graphics[
{
{Thickness[.006], Circle[{0, 0}, r]},
{PointSize[.03], Red, Point[r {Cos@#, Sin@#} &@t]}
},
PlotRange -> 2
],
{t, 0, 2 \[Pi], Animator},
{r, 1, 2}
]


Here we have a constantly moving point while still being able to manipulate the radius of the circle. (Granted the gif is doesn't exactly represent this since it was made with Table, but you get the idea)

Whereas, if this was done with Animate, all variables would evolve at the same time.