Great question! The answer is… it depends. In short, AO helps pop out details by adding soft shadows to crevices. Cycles and Eevee AO (the AO pass) works based on the geometry in the scene. An AO map, however, gives the same effect but since it’s a texture it can also be used for details that are too small to be represented by the low poly geometry.
Example of a Cycles material without an AO pass but natural self shadowing because of the raytracing:

Added AO map for slight extra pop:
Not a huge difference. Eevee without any AO:

Eevee with AO pass:

Eevee with AO map multiplying the base color in the material but no AO pass:

Eevee with AO pass and AO texture:

So, do you need an AO map in Cycles? No.
Might it help spice things up if the scene is looking too flat? Sure!
Do you need an AO map in Eevee or a game engine?
Same answer as for Cycles if rendering with screen space AO, except it may also help in situations where screen space solutions fail (example here: https://blog.cgcookie.com/blender-cycles-vs-eevee-15-limitations-of-real-time-rendering/ ).