I think this is a 2-part question.
1. Do you do your taxes by hand without software? I tried that a few years ago with a State return for my state of residence (had downloaded the version for my employment state and didn't want to pay for a second one). I'm an actuary, math-friendly, legal-ese friendly, Excel-friendly. Still too many "mini-worksheets" and twists and turns and I made a few mistakes, which the state's computer flagged. I owed more $$, of course. Now I use software. Always.
2. Do you e-file or send a paper copy? I'm old-fashioned. I send a paper copy. I just figure I'll make them work for the details if they want them rather than giving them data to mine. That's delayed refunds in years when I overpaid, but I'm willing to wait. Never had a problem with returns going astray.
Yeah, I'm not clear whether the OP is talking about doing 1 or 2. I assume since the cost of TT is mentioned, it is 1. I won't ever do it by hand anymore now that I've used TT or similar for a number of years. Reasons:
1) Math errors. I can use a calculator but there's still the chance of punching in a wrong number or transposing numbers.
2) Filling out the forms wrong. As I recall the LTCG and QDivs worksheet is somewhat complex in what they are asking for. Or I might forget to carry over capital losses, which tax programs do automatically. Handling foreign taxes paid doesn't look simple. I'm sure there are more things I can do wrong.
3) Making changes. If I find another 1099-INT or charitable deduction, no worries with a tax program. I just punch in the info, and everything gets updated. On a paper copy, I likely have to redo the affected form, the main 1040, the CG/Divs worksheet, and recalculate the tax.
4) Running what-if scenarios. Tax programs are great for running scenarios on what happens if I convert more of my IRA to a Roth, or other such things.
5) Retrieving info on past returns. I keep all my returns on my laptop, backed up. If I need to look up anything, I just open the .pdf file, and there it is. No going to the file cabinet to find the right file folder. My thumb drive for backup data also takes up less space than past returns in my fire proof safe.
6) Time. I'm sure doing it by hand takes a lot more time, and even though I'm retired my time is worth something, especially since I never liked filling out tax forms by hand.
7) Amended returns. Pretty simple with tax programs, more work by hand.
8) 8. I forget what 8 was for. I'm sure there are more reasons.
If it's #2 above, as has already been said, e-filing fed taxes is free so I'm not going to spend my ink printing out forms to mail in, plus wonder if they actually got there. I know when my e-filed form has been received. State taxes do have an e-file free in my state, and I think it's just 2 pages to print, so I do print and mail those, reluctantly.
So, since I no longer can get free Turbo Tax (Deluxe), what's the downside to paper filing? Obviously, there's the delay in getting any refund. But is that delay much longer? 
