Probably a good bit. I went from a 910D2 to a TSR3 a saw a huge difference. The performance of the head is tons better and the new shafts are so much better than the one on my 910 and probably the one on your 915 also. My TSR2+ 3 wood actually goes as far as my old driver did. Go get fit and you won't regret it.

Technology is markedly improved. Off center hits will be vastly improved. The key will be getting the correct loft and shaft suited to tour swing. After 915/917, Titleist has worked hard in R&D to compete on distance.

You could realistically expect a noticeable improvement in dispersion, especially off-center hits, when moving from a 915D2 to a TSR2 — but the exact gain depends on a few factors like shaft fit, swing consistency, and strike location.

Key Differences
Face Tech: The 915D2 uses an older Active Recoil Channel, while the TSR2 has a thinner, more responsive multi-plateau face design. The result? TSR2 maintains ball speed better across the face.
MOI / Forgiveness: TSR2 is more forgiving heel-to-toe and higher-launching. You'll likely see tighter dispersion on mishits.
Aerodynamics: TSR2 is more aerodynamic, helping swing speed consistency and strike location, especially for faster swingers.

Expected Improvement in Dispersion
Typical gains in left-right dispersion: tighter, especially on heel/toe strikes
Launch/spin control is more stable shot-to-shot, meaning fewer big misses high or low
More consistent carry yardages across the face

You won’t suddenly hit every fairway, but going from a 915D2 to a TSR2 could absolutely help you tighten your shot pattern, add a bit of ball speed, and give you more confidence on the tee.

I would not be set on a TSR2 as optimal improvements cannot be had by buying an off-the-rack driver. Going to a fitter could result in better performance improvement through a different model...maybe even a different brand! (Which would be highly unfortunate for you! Haha)