How can I display my maps (from maps.google.com) on Android smartphone? How can I access to "my maps" saved from desktop website (maps.google.com) from my mobile device?

When I use maps application on my device I am not able to find my saved maps with my POI and my settings

I am using Android 4.4 on Nexus 4
Have you explored the layers option in your Maps app? Your saved maps can be shown as a layer.

In the following image (courtsey: HowTo.Cnet.com) you can see the "My maps" section. Expanding My Maps will show all your saved maps. It requires a data connection to use this feature.


Update:
While composing this reply, I cross checked my input with the Google Maps app in my mobile. My mobile is running v2.3.3 and having Google Maps of version 6.x. I could see the "My maps" section in the layer and I could overlay one of my saved maps. 

When I saw OP's comment that "My maps" was not found, I cross checked it with my Nexus 7 running 4.3.3 and Google Maps v7.3.3. It has indeed no "My maps" at all. After some search, I understood from this page that "My Maps" has been removed - may be in favour of Google Maps for Business. 

Update:
I just found a new app from Google at its Play store named "Google maps engine" which seems to overlay the stored (though not the old ones, but only the ones that were created with new Google maps for desktop) maps.