A private terrace is outdoor space you do not share. It is not the roof deck, and it is not a reason to invent a dollar premium. Seventeen listings on this site mention a terrace or a balcony. The rest do not have one.

What you actually get
You get a place to stand outside without going down to the street. In warm months that is a chair, a morning coffee, and somewhere for a plant that is not a windowsill. In January it is a door you keep closed. Murray Hill is not a climate where a terrace is a second room for half the year. If you are paying more for one, price the months you will use it, not the photograph.
Wind is the part the photo leaves out. A high-floor terrace on an avenue is a different object from a recessed balcony on a side street. Furniture that is light enough to carry up is also light enough to move in a gust. Ask to step outside on the tour, including on a day that is not still.
What it is not
The building's roof deck is a shared amenity. Every resident can use it, subject to the building's rules for that space. A private terrace belongs to the apartment. Touring the roof deck does not tell you whether the unit you are signing has outdoor space. Read the listing. If it does not say terrace, it does not have one.
A terrace is also not free storage, a grill station, or a place to leave anything the lease does not allow. Published pages here do not set a terrace rule for furniture, planting, or open flame. Ask before you buy a chair, and do not assume a propane grill is permitted. Propane heaters are illegal in the city regardless of what a lease says.
How to weigh it
Stand in the living room and then stand outside. If the terrace is the reason you want the apartment, it should still be the reason when you picture it in November. If you want outdoor space a few evenings a month, the roof deck may be the honest amenity and the apartment without a terrace may be the better rent. No public page here states a terrace premium in dollars. Do not let a listing site invent one for you.
Filter listings for the ones that mention a terrace, and tour that unit, not a similar one. Call (212) 684-3222 if you want to know which available residences have outdoor space this week.
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