Do you think using Photoshop > threshold is same as using image tracing in Adobe illustrator?

No, threshold in Photoshop is different. For tracing an image in illustrator makes vector and there is a huge difference in vector and raster image.

No, Its different. Threshold in only on bitmap but tracing is for vector. 

It's not the same in Photoshop you only darken the pixels but in illustrator you make a vector of the pixel