My machine was updated today to Ubuntu 16.04.2. However, the kernel is still "4.4.0-62-generic". What happened (it seemed that the principal advantage of the 16.04.2 version was the update of the kernel to 4.8) ? Do I have to make a fresh install ?
"Updated to 16.04.2" and "installed 16.04.2" do not render the same result. The HWE packs for the Kernel and X aren't automatically installed onto existing non-point-release installs of Ubuntu.
For the main reason that it'd really annoy a whole load of people who have a stable system one day and then have to battle a new kernel. That's why the .0 kernel gets a full five year support.
You can opt-in to install the HWE packs (and have been able to for a little while) with:
sudo apt-get install --install-recommends xserver-xorg-hwe-16.04
There is an important point to make here that starting from 16.04.2, point releases will be on rolling HWE stacks. You will automatically be shunted onto newer major versions of X and the Kernel.
This is a really important thing to note if you're deploying Ubuntu in low-maintenance situations. You don't want to climb a 50-foot ladder to get to your digital signage controller because a dodgy kernel update caught you off guard.
There is no "principal advantage" of installing the 4.8 kernel.
In most cases it is not necessary, and it is more convenient to keep the original LTS kernel. You may need to update the kernel if some hardware is not supported by the 4.4 kernel.
That is why kernels do not automatically update to another major version.
Anyway, you can always install the 4.8 kernel by running
sudo apt install linux-generic-hwe-16.04
The 4.8 kernel is installed if you do a fresh installation from the 16.04.2 Ubuntu iso.