I got my BR-V S 3 weeks ago. Sold my 1.6 Altis 2016 to get this car, so this is sort of a comparison. Overall there are a few things that, for me, make owning a BR-V a very different experience.
When it comes to driving experience, 1.6 Altis has a 4-speed automatic transmission, which is extremely 'jhatka' prone when accelerating. Its like you can drive the corolla only in 2 modes. 1) at very slow acceleration where you're just maintaining your speed. 2) you're tearing through roads like a mad man with a lot of engine rev-ing sound. The transmission of 1.6 automatic corolla is really really bad in terms of driving experience. The CVT is really really smooth and works like a charm on the BRV. Man it's even better than the Prosmatec on 8th-gen civic (owned previously). However, mechanically speaking, Corolla's transmission is more durable compared to CVT both on Altis 1.8 and BR-V (if you change gear oil on time nothing happens to either).
Corolla had less cabin noise and smoother suspension, that said the suspension on BR-V is very good and many folds better and less noisy than 8th and 9th gen civic(previously owned). Haven't driven the 10th gen civic enough to comment.
As far as the interior is concerned, I love the meter colors and the air conditioning in the BR-V, especially the rear vents is a nice usable touch IMHO. Corolla has a good AC too but the steering looks like it was bought from a 'ghubarey wala' at best. The meter was. just. pathetic! (they upgraded it in the latest 1.6) Interior in BRV is much better than the 2016 Altis I owned. The in-dash multipurpose entertainment system is tastefully made and has a barrage of features (includes navigation by default unlike Toyota). If your phone was connected over Bluetooth to the in dash system in the Corolla Altis 1.6, the only available ringtone of the in-dash system was too loud (volume couldn't be decreased), could not be disabled and it would make the ears bleed of anyone sitting in the car.
The driving seat of BRV is much more comfortable for long drives, the angle of the steering to the driver is much more user friendly. In the Altis the angle of the steering to the driver is more like what you'd see on a Bedford Truck (an exaggeration to give a sense). IMO civics have the best arrangement in this regard.
I get 11.5 Km/liter in BRV, in Corolla 1.6 Altis I got 10.7km/l under same conditions.
These are my 2 cents where I just covered some things that are relevant when driving everyday. Fortunately I owned these 2 cars in succession hence the comparison. Detailed comparisons and reviews can be found in other posts. Cheers!