An uncommon Medallion of Thoughts lets you cast detect thoughts up to three times a day, while a rare Potion of Mind Reading is a one-use item.
Are there any mechanical advantages to using the more expensive potion over the cheaper medallion?
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There are two components: Attunement and Concentration.
Attunement is obvious. You have 3 attunement slots to share between all the magic items you may own. The Medallion of Thoughts uses one of those--a potion does not.
The other component is Concentration
While wearing it, you can use an action and expend 1 charge to cast the detect thoughts spell (save DC 13) from it
Some magic items allow the user to cast a spell from the item, often by expending charges from it. The spell is cast at the lowest possible spell and caster level, doesn’t expend any of the user’s spell slots, and requires no components unless the item’s description says otherwise. The spell uses its normal casting time, range, and duration, and the user of the item must concentrate if the spell requires concentration.
However, the Potion of Mind Reading simply causes you to gain the effects of the Detect Thoughts spell without actually casting it.
When you drink this potion, you gain the effect of the detect thoughts spell (save DC 13).
As a result, the Potion does not require Concentration in order to use. If you're using the Medallion of Thoughts, then you have to use your Concentration to maintain the spell--so you can't have any other Concentration spells running at the same time. If you use the Potion, then it doesn't consume your Concentration and you may use another spell alongside it.
The ability to effectively stack Concentration spells is potentially quite powerful. This particular one allows you to do things like stacking Calm Emotions and Detect Thoughts to get someone to chill out enough that you can read their mind. Or Bane along with Detect Thoughts to reduce their chance of making the Save. Or Infernal Calling/Summon Demon with Detect Thoughts to allow you to call up a demon and interrogate it by mind-ripping it. Even at the most basic...it makes Detect Thoughts potentially viable in combat...letting you read the mind of the Wizard you're fighting so you can anticipate the spells he's about to cast--while not interfering with your ability to fight effectively.
The Medallion of Thoughts requires attunement. A Potion of Mind Reading doesn't (potions never do, as far as I have found).
While we can't know for sure if this is a factor, items that do not require attunement are more immediately "useful" in the sense that they can just be picked up by anybody and used straight away. Conversely, an item that requires attunement has a built-in limitation - a character must attune it first which takes time, only that character can then use it and you can only have three attuned items.
Despite the weak correlation between rarity and "power", attunement requirement certainly has a role in both aspects.
The potion does not require attunement or concentration, which is nice, but it is still a single use item.
The advantages are negligible compared to this one, huge drawback.