Basically they are not same.

Without soil: you can grow plants with other support while providing water and nutrients (such as lattice grown on water with nutritiants).

Aerobic agriculture it is not necessory to be without soil. For example we can grow rice with minimum tillage.

Aerobic agriculture: is now used commonly for rice cultivation with no standing water. Just like any other field crop. It basically evolved as a technique to conserve water because huge amount of water is used to grow rice sometimes as high as 5000 litre water to produce just a kilogram of rice.

There can never be agriculture without soil. Only on a limited scale, it is possible. This is commonly referred to as hydroponics. Plants grow in a nutrient solution and their nutrient requirements are met through the solution.

Thus, both the systems you queried are different.

@ Sakir, dr. Desouza, is correctly explained the aerobic agriculture, the term which is commonly used in rice cultivation without standing water. Agriculture without soil is the hydrophonic technique using specially designed green house and rooftop. It is completely different from aerobic agriculture where measured amount of required nutrients is dissolved in water to grow specific plants.

I agree with Desouza's answer.

Dear Shakir Msser , I think your question is implying a confusion between aeropic agriculture and aeroponic agriculture. our colleagues have explained what aeropic agriculture is. on the other hand, aeroponic agriculture is growing the plants in a system where its roots are suspended in a dark space where water and nutrients are sprayed on those roots. aeroponic is a type of agriculture without a soil or what is commonly called "soilless culture"

I agree with the answer of Youssef Sassine

Recently aerobic agriculture is developed for rice where in rice farming is done under water limited conditions,whereas in soilless agriculture i.e.hydroponics crop are grown in nutrients solutions.

Following!