Are equivalent positions really equivalent?

The following question from a reader leaves me puzzled:

"You bring up an interesting point and something that I read in the book
that I did not completely understand – I did think about what you said
about married puts being equivalent to a call (synthetic) – but even though in
an equation that may be equal, isn't it different in real life?"

As I said; puzzled:  true in the equation, but different in real life?

I'd appreciate some feedback.  When you see an equation as 'proof' that something is true – do you doubt that it's true in real life?  In other words, is the proof meaningless?

When I've been telling you that writing a covered call is not different from selling a cash-secured put, do you believe that to be true?  Or do you believe that either one of these strategies is better than the alternative?  I'm not talking about commissionsI'd like to know if anyone believes these strategies produce different financial results – after dividends and interest considerations are included. 

Do any of you use one strategy instead of the other because you believe it makes more money?  I certainly understand if you use one because you feel it's easier to trade or your broker doesn't allow both.

Along the same lines, do those of you who use collars truly believe that the position gives the same results as selling a put spread or buying a call spread?  I know it true.  But do any readers have doubts?

The problem for me is that I know that equivalent positions are truly equivalent.  When the options are not mis-priced (which is essentially all the time), when dividends and interest costs are included in the calculations, profit/losses are equal, no matter which of the equivalent positions you trade. 

In the real world, some positions are easier to enter and/or exit.  That may be a good reason to prefer trading one, rather than the other.  But isn't that decision is based on convenience, reduced commissions, or some practical consideration other than believing that one is just more profitable than the other?  

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