When will playing Rubber Bridge vs. Duplicate Bridge affect your decisions?
I wholly agree with thesun's answer, particularly "you must squeeze every ounce of potential out of your cards".  But since you've posed particular queries at the end of your question:

Part-scores are the most obvious difference.  If you have strong Hearts, but not strong enough for slam, a Rubber player looks at the scoresheet; with 60 points below the line, there is no point going above 2H unless forced (overtricks score the same as bid tricks, and there is no risk of failure).  A Duplicate player knows there is no line: bid to 4H if you can make it.  Similarly with honours: a marginal hand with KQJT, worth bidding at Rubber because of the cushion 100 honours will give, is not worth bidding at Duplicate.

Strategy: again, Rubber bridge is about making the best of your situation, while Duplicate is about making the best of your cards, Take sacrifices: a Duplicate player knows that if he goes one down doubled but stops opponents making a game, he has done well out of that hand.  At Rubber bridge, you have saved the rubber for now, but if the opponents win a few hands later, all your sacrifice achieved was to cost you a few hundred points.  More generally, if by strategy you mean scoring as many points as possible (as I think you should), any change in the scoring method will affect your strategy.

Usually, you can't tell under which system a particular hand was played; obviously, if one side started with 30 below the line it was at Rubber, and if two declarers took different routes it was Duplicate.

And if you are really a good Rubber bridge player, you will be (or are) a good Duplicate player.  'Good', of course, is compared to 20 or 30 others, rather than 3, and you may find that your system is too ill-defined to be acceptable without work: but it's the same game, and just as enjoyable if not more so.