Sovan,
The ArcGIS License Manager is backward compatible so we always recommend you use the latest version of the license manager even if your desktop application remain the same. On that note, let's take a look at your immediate questions:
1. Can we use the same license manager for ArcGIS Desktop 10.6.1 or do we need to upgrade to License Manager 2018.0?
answer: No, you must upgrade to the same or later version as your Desktop application. There are upgrades to the core application that also need the same or later upgrade in the license manager.
2. What are pros and cons for continuing with 10.1 -10.6 license manager and upgrading to 2018.0?
answer: ArcGIS License Manager upgrade is an accumulation of features from the previous version so there are numerous advantages. This is why it is backward compatible for any version of ArcGIS Desktop 10.1 - latest version. The disadvantage is that fact someone have to install the later version.
3. What are the difference between both license managers?
answer: Not much change for the user. On the surface, there's little change. You may notice a few more extensions or version change for ArcGIS Desktop and Pro. The change is under the hood where we updated the Flexnet Publishing tool.
