Question: It seems that any modification you can do to a weapon on the Item Modification Table can also be done from Inventory.

Why are there so many tables and why would I want to use them?

Answer: 
UPDATE: New features to the game including customizing augments. This feature requires the Item Modification Tables now. See Matthew Read's answer. The original answer that applies to the game as it was originally released, is still included here:

ORIGINAL: 

It is a remnant from earlier item modification game mechanics Bioware was trying. 

Currently the Item Modification Table has no special purpose. You can modify an already-modifiable item by `Ctrl + Right Mouse Click`-ing the item in your character or inventory window. You can still use the table to start this process if you want, but you don't need to.

EDIT: One on each starter planet is also used as the target of a "how to use item mods" intro quest.
The Item Modification Stations are now required to add Augments to your gear. When you buy an Augmentation Kit you have to take it to one of the stations and pay to apply it to your gear, giving you an augment slot. You can then use the station to add an augment or simply modify the gear as normal with Ctrl + Right Click. (They still allow you to adjust other mods with the stations.)

Augments are exactly the same as other mods, they add extra stats. All equippable items can be augmented.