Str 18 with a +1 long spear, BAB +7, power attack and studied combat +3 comes to attack +13, 1d8+16 damage. 2d8+16 w/vital strike, 2d8+32 w/mythic vital strike.
Str 10, Dex 18 with a +1 rapier, BAB +7, fencing grace and studied combat +3 comes to attack +15, 1d6+8 damage.
Yes Str-based does a lot more damage. There are some tricks which work only with a rapier - a 1-level dip in inspired blade swashbuckler is great for your survivability for example, or if one of your talents is mutagen that works much better as a Dex mutagen than a Str mutagen for an investigator due to where the penalties fall, and having a hand free for a swordmaster's flair is nice. Also Dex skills are probably more important to you than Str skills, and an investigator is the king of skills.
Actually in a mythic campaign STR builds don’t necessarily do a lot more damage. Mythic weapon finesse allows you to use DEX to damage on any weapon that can be used with weapon finesse including two handed weapons. This means you can get 1.5 DEX to damage. If you have at least a 13 STR you can also get +3 to damage per -1 when using the weapon two handed by taking power attack. The perfect weapon for this is the elven curve blade which also has a crit range of 18-20.
One thing to consider is that as an investigator you only have proficiency with light armor. Most STR based builds rely on heavy armor. That means if you go STR based your AC will probably be very poor.
vital strike is probably not worth it on either build. An investigator is not a full BAB class so it takes a lot longer to qualify for the feats. At 10th level you only qualify for vital strike and will not qualify for improved vital strike until 15th level, and never get access to greater vital strike.
To be honest, with Mythic Vital Strike the stat you use for damage is a lot less important than the number of weapon damage dice you get. Especially since you will be using only Vital Strike for most of your build, since you won't qualify for improved vital strike until level 15.
For example, let's say you were using a butchering axe, and your strength was an 18. That would mean your weapon's damage dice (multiplied by 2) would be 6d6, so we would multiply all additional damage added to the total by 6. That means your strength would add a total of 24 damage to the total, and if you were power attacking, power attack would add 36.
6d6+6(4)+6(6)= 6d6+60
With the "Impact" enchantment, the effective size of your butchering axe increases by 1, bringing its damage dice to 4d4, and 8d4 with power attack
8d6+8(4)+8(6)= 8d6+80
The fastest way I can think of to get more weapon dice is the Shikigami Style line of feats, which actually sounds like a great option. What's better for a "thinking man's" combat style than using whatever is around you to your advantage against your enemies? Also, hard-bitten detectives (like many the roles of Humphrey Bogart) get into scrapes and have to fight with improvised weapons all the time. There's nothing quite like knowing that you're more dangerous with a broken bottle than most men are with a greatsword.
Wielding a sledgehammer with all three Shikigami Style feats means that you will be swinging for 6d6, or 12d6 with Vital Strike. If you use a Traveler's Any-Tool transformed into a sledgehammer, it will also be a +2 weapon.
That makes the damage into:
12d6+12(4)+12(6)+12(2)=12d6+144
Also, the Surprise Weapon trait gives a flat +2 bonus on all to-hit rolls with improvised weapons, which is like getting Weapon Focus twice for the cost of a trait. Very useful.
Best of luck!
UE, that is not at all how mythic vital strike works. A weapons damage dice refers to the whole weapon's damage dice not to each die individually. A butchering axe does 3d6 + mods damage, 6d6 + mods with vital strike, 6d6 + 2*mods with mythic vital strike.