In general, I would always advocate erring on the side of caution in these situations, but that should not mean a blanket "don't do it". I certainly would not advise that here. Looking at the Release Notes is the first place to start when trying to decide if there are any advantages to upgrading, like corrected anomalies, or any disadvantages, like an anomaly that may appear as a result of upgrading. But they won't always have poked their heads up before the Release Notes have been issued...
Also, searching the AB Knowledgebase for issues with specific releases is best advised...
620043 - 1756-ENBT or 1768-ENBT interrupted communications to a remote device could cause degraded bridge performance
Access Level: Everyone
That anomaly starts at revision 6.002 specifically and persists right up to the latest revision 6.006. You have a 1756-ENBT running stable in this system using revision 6.001. So that issue is "one" example where I would be cautious. I'm not going to exhaustively search and decipher technotes on all of the revisions above 6.001 for you. That was just to guide you in how to approach these upgrades.
If you review the other releases since your oldest revision 3.003, you will see there are several corrections to network related vulnerabilities and bugs, such as potential Denial of Service (DOS) attacks. So there may indeed be Security benefits in upgrading.
If you do decide to upgrade, then being prepared to "roll-back", just in case, is probably the best advice here, or having a spare module at the older firmware is a plus, as mentioned.
Just one "small" thing to note if going to the latest: To select revision 6.006 under a 1756-ENBT module profile, you must be using RSLogix v17 or higher. Else use Compatible Keying. That's "just in case" this application/site stipulates "Exact Match" for all modules and your project happens to be that little bit older.
Regards,
George