Lets assume 2 people in the same scenario.

Both people :
make $100,000 a year (assume this is constant)
put away 15% toward a 401k
same company match so ignore that for now.
have the same funds and on average get back about a 7% return (is that a realistic average?)
will retire in 15 years


The only difference is Person 1 is doing Pre-tax contribution and Person 2 is doing Post-tax contribution.

At the end of 15 years what is Person's 1 total amount.

At the end of 15 years what is Person's 2 non-taxable amount and taxable amount.

No idea what formula to use to calculate this.

If they both put in 15% then they are both contributing $15,000.00
If they both make the same rate of return then they both have the same balance, about $396K

The difference isn't in the future balance.  The difference is in the cost in the current year.
It will cost ~$3,750 more per year in (Federal alone) taxes to make the Roth contributions.

/answer

Bonus info:
The person making traditional contributions would be able to make a maximum contribution of $18,000/ yr (2015 max) and still have the same paycheck because of the tax savings.
Following that theory, the additional $3,000 each year is also earning and compounding (I'm not going to calculate this scenario since you didn't mention it as an option).

Also, assuming this is the person's sole source of taxable retirement income a single person contributing to a Roth account would have given up at least $10,300 (2014 numbers) in tax free income from the standard deduction and personal exemption at the time of withdrawal.

I didn't follow the part about both contributing $15k.
If person one is pre-tax that makes sense.
If person two is post-tax then isn't it 15% of their post-tax amount which may be something like $11,250.

You said that the both make $100k and would contribute 15%.  100,000 X 15% = 15,000
The 15% is calculated from the Gross Income (100,000), so it's the same for both.
If you put 15% in to a Roth account you put 15% in, but your tax number will also be higher, making your paycheck lower than the guy putting 15% in to a traditional account.  The difference will be taxes.
You can't say "I am putting 15% in to my Roth 401k, but really it's 12% to the 401k and 3% towards the taxes on that amount".
You can't calculate it in reverse.

Does that make sense?

Try this calculator, it's along the line you seek.
https://www.schwabplan.com/download/RothCalc/RothCalculator.htm

Plus this being a good overall explanation of the two:
https://www.fidelity.com/viewpoints/retirement/roth-or-non-roth-401k

But the key difference is :
The only difference is Person 1 is doing Pre-tax contribution and Person 2 is doing Post-tax contribution.

If one person is doing Pre-tax and the other Post-tax the 15% amount should differ.
Pre-tax would be 15% of 100,000
Post tax would be 15% of probably 80,000.

I will check those out, but I was able to get some numbers.
The pre-tax person ends up with :
$264,589 from salary + $52,918 from company match for a total about $317,507 and about $500K when including a 7% rate of return.

The post tax person ends up with :
$198,442  from salary + $39,688 from company match for a total about $238,130 and about $400K when including a 7% rate of return.

With a 30% tax on the $264,589 which is a bit high it appears the Pre-tax scenario still comes out ahead.
Even more so with a smaller tax bracket.

Remember that you need to pay taxes on total amount when you withdraw on the pretax.  Qualified withdrawals on the post tax are not taxed.

Pre tax: $817k but you will be taxed when you withdraw.

Post tax:  $638k but you will not be taxed when you withdraw.

Depending on your tax bracket (keep in mind taxes rarely go down) you will likely be better off post tax.  Having some in both is also a good option for tax diversification.  Don't get caught up in $$ amount until it reaches your pocket and you pay the tax man.

If you are looking at Roth vs. non-Roth retirement savings, you also need to look at how much is available, after tax in retirement, and not just the amount of the wealth accumulation in the two accounts.

Did some further research and found the following tax rate for withdraws :
up to $18,150 ----- 10%
$18,151  to  $73,800 ----- 15%  
$73.801  to  $148,850 ----- 25%
$148,851  to  $226,850  ----- 28%

If Social Security is still around I can see withdrawing in the range of 15% tax which makes the Pre-tax the better option.
If Social Security is not around I can see being in the 25% which makes the post-tax the better option.

The above is based on married filing jointly or qualifying widow/widower.
The amounts for a single filer is significantly lower meaning the range $36,901  to  $89,350 is taxed at  25%

Duh, after further research I see what you mean.
Originally I thought the X% was taken from the Net amount after taxes, but that assumption was wrong.
Whether the person is doing pre or post tax the amount is the same.  Now I get it.