Version of SSMS won't be making the difference. If the Source Database (from which the backup was made) was using a newer version of SQL than the server which the Target Database is on then a Restore is not possible
Should be possible to use RESTORE HEADERONLY [on the backup file] to see what the version of SQL was running on that server when the backup was made
12.0.2269.0 is SQL Server 2014 RTM (MS15-058: GDR Security Update July 14, 2015)
I have no idea what 13.0.15900.1 is.
13.0.1601.5 was the first release of SQL Server 2016 RTM, there are higher build numbers for updates to that ...
13.0.1400.361 was the last Release Candidate, prior to release, that I know of
13.0.15900.1 seems to fall in the middle. Perhaps some patched version of a release candidate for SQL 2016?
Are you sure that is the build version of the BACKUP File (rather than the version of SSMS?) [Although it doesn't feature on any version number list I have for SSMS - the latest 16.4 dated Sep-2016, that I have seed is 13.0.15800.18)
What I have done at my SQL server management studio is updated using "check for updates" and got this version as noted. anyway, what I don;t understand is why I have the engine in one version while the management studio is on other, although I have done an update.