Strong Foes in Super Mario RPG. The guide explains how to spawn and identify strong foes, their EXP and Frog coins you can get, and how they differ from normal enemies.
How to Spawn and Identify Strong Foes
They Appear Randomly

Strong Foes appear randomly when you engage an enemy on the field. You can identify them by their name, which will be yellow in battle with the word "Strong Foe" written on it.
You Won't Know Until You Fight

Strong Foes have no change in their field appearance. Therefore, you can't identify them on the field, and you won't know if it's a Strong Foe until you start the battle.
You Can't Intentionally Fight Them
There is no way to intentionally make Strong Foes appear or guarantee a fight with one. To encounter a Strong Foe, you have no choice but to fight enemies repeatedly and leave it up to chance.
Benefits of Fighting Strong Foes
Increased EXP and Coins

When you win a battle against a Strong Foe, the amount of EXP and coins you get is doubled. However, this only applies to the Strong Foe; the EXP and coins from other enemies in the battle will not increase.
You Can Get Frog Coins

In battles with Strong Foes, you will receive one Frog Coin in addition to EXP, coins, and dropped items. The appearance rate of Strong Foes isn't very high, but they can appear even early in the game, so it's possible to farm Frog Coins by defeating many weak enemies.
Yoshi's Cookie is Ineffective Against Strong Foes

Yoshi's Cookie, which turns monsters into items, is ineffective against Strong Foes. Therefore, be aware that you cannot use Yoshi's Cookies to deal with them.