Now comes the hard part.
They each give themselves an M-year annuity (withdrawing from their portfolio so that there's nothing left after M years).
These annual annuities are:
Sam:  S₁ = P₁ R₁/{1-(1+R₁)^-M}
Sally: S₂ = P₂ R₂/{1-(1+R₂)^-M}

In retirement, Sam and Sally are each in a tax bracket of T₂.
Sam pays taxes at this rate on his entire annuity, so he's left with:
S₁(after tax) = P₁ R₁/{1-(1+R₁)^-M}(1-T₂)
and if we substitute P₁ we get:

Sam's After-tax Income:  S1(after tax) = A{(1+R1)N-1}/{1-(1+R1)-M}(1-T2)

Sally, on the other hand, pays taxes at a reduced rate ... and only on her gains.
So what are her gains?

We go slowly here ...
Her portfolio P₁ (see above) has a nontaxable component, namely all the after-tax dollars she contributed:
N years at A(1-T₁) per year gives:
nontaxable dollars (NTD) = NA(1-T₁).
The taxable part is what's left (they are her gains), namely
taxable dollars (TD) = P₁ - NA(1-T₁).
From the "nontaxable dollars" of $NTD she gets an M-year annuity of: (NTD) R₂/{1-(1+R₂)^-M}
From the "taxable dollars" of $TD she gets an M-year annuity of: (TD) R₂/{1-(1+R₂)^-M}
The sum is, of course, her total annual annuity which we can write as:
Sally's TOTAL annuity = A(1-T₁)[ {(1+R₂)^N-1}/{1-(1+R₂)^-M} ] ... but what taxes does she pay?
The "nontaxable" part of this total income is just $NTD, spread over M years, so it's NA(1-T₁)/M.
Hence (and therefore), we'll let her pay full taxes on her total income then give her back the taxes on this "nontaxable" part:

Sally's After-tax Income: S2(after tax) = A(1-T1)[ {(1+R2)N-1}/{1-(1+R2)-M}(1-(3/4)T2) + (3/4)T2N/M ]

Note: we've assumed her gains are capital gains which accounts for the factor (3/4).
If'n it ain't capital, then y'all kin change the (3/4).

I'n y'all kin get a better return OUTside ... then ... the "miracle of compound interest" ...

Check this out:

DOW vs TSE

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P.S. If'n ya got Excel, there's a .ZIP'd spreadsheet y'all kin download ... to play with: RRSP: IN_or_OUT?

... ain't Math wunnerful?

To get a feel for how much more you'll need (in terms of Return on Investments) in order that investing OUTside gives y'all more money than investing INside, consider this approximate, quick-and-dirty calculation:

Suppose Sam begins with $A and, after umpteen years of investing INside, this money grows by a factor F, so his portfolio is now AF. He cashes it in, paying taxes at the rate T (for a 44% tax rate, we put T=0.44), and is left with:

Sally, on the other hand, invests what's left of the $A after she pays taxes at the rate T, namely A(1-T), and, after umpteen years, we suppose it grows by a factor G. She now has a portfolio worth A(1-T)G. She then cashes in, paying taxes at the rate (3/4)T ('cause her capital gains are taxed at a lower rate), leaving her with:

Does Sally have more after-tax money? She will, if A(1-T)G(1-3T/4) > AF(1-T), that is, if

Moral? Provided she gets a sufficiently bigger return on her OUTside investments, she'll come out ahead ... after taxes. For example, if Sam's return is 10% per year for 30 years, then his gain is F = 1.10³⁰ = 17.4 and if the tax rate T corresponds to 44%, then T = 0.44 so F/(1-3T/4) = 17.4/(1-.33) = 26.0 and we conclude that Sally's thirty-year gain should exceed 26.0 in order to beat ol' Sam. Sounds a bunch, eh? However, it corresponds to 26.0^1/30 - 1 = .115 or 11.5% per year^* (roughly).

It doesn't sound like much of an increase over INside investment returns ... but what are the chances of doing that? Slim?

On the other hand, if taxes on Capital Gains were at 50% (instead of 75%) ... and taxes were 44% then we'd get (changing the numbers, above): F/(1-T/2) = 17.4/(1-.22) = 22.3 and 22.31/30 - 1 = .109 or 10.9% and maybe that's easier to achieve, eh (especially if'n y'all don't sell nothin', so don't pay annually on capital gains). For a tax at 50% of Capital Gains, this quick and dirty approximation illustrates that if you're in a lower tax bracket and you're investing over a looong time ... then maybe OUTSIDE is better (if you can achieve the larger after-tax returns)

Of course, Sally doesn't really pay taxes on all of her portfolio - some of it is after-tax contributions. Further, the tax rates during the investing phase needn't be the same T and during the withdrawal phase. Further, who (in their right mind) would cash in their portfolio all at once - which explains why we considered buying an annuity, above. Anyway, it gives some insight - perhaps - and that's why we're calling this quick-and-dirty.

Maybe it's mostly dirty ...

^* In general, the Quick-and-Dirty formula is like so:
If the INSIDE return is I (example: I = 0.10 meaning 10%), then the OUTSIDE return should exceed (1+I)/(1-gT)^1/N - 1 where T is the Tax Rate (example: T = 0.40, meaning 40%) and g = taxable portion of Capital Gains (example: g = 0.5 meaning 50%) and N = Number of years of investing (example: N = 30 years).

For I = 0.10 (10% return) and g = 0.50 (Cap. gains taxed at 50%), the Quick-and-Dirty extra return required (over and above the INSIDE return) is 1.10/(1-T/2)^1/N - 1.10 and depends upon N and T ... look like so:

See also the Calculator