I'll chime in as someone who started on the Free Weekend.  Several game functions are blocked/missing in the tutorial, and did indeed cause a lot of confusion until I won and started a new "Normal" game with the full features.  Off the top of my head:

• Fleet options are extremely limited. No Defend, Auto-explore, etc. options are available.  As mentioned, I could not complete the Photobomber quest, and had issues with pirates always slipping through my systems as I could not block them even with my fleets there.

• Some politics features are not there. For example, the Federation government type does not get its +1 colonized system bonus in the tutorial, and thus appears simply inferior to the other options.

• Several Battle Tactics cards are locked or missing. Normally you start the game with the "Turtle" tactic card which allows for short range fights, but in the tutorial it is simply not there and I had trouble find any tactics cards with the middle lane on short range.  As a result I was confused that  Ihad chosen to outfit my ships with Kinetic slugs, but until I unlocked the 2nd and 3rd flotilla, I had no access to any Battle Tactics which put my ships into close-range.