You are right NOT to Format the drive. In fact, I suggest you should NOT even try to WRITE anything to this drive with the set-up you have.
I would suggest you check into power first. An HDD in an enclosure with insufficient power will not respond properly, and MAY partially respond by writing corrupted data. So check the power before going any further. ALSO see my note #2 at the end to be sure the HDD actually works.
The starting point here is that any USB2 port (I'm assuming you are not using a USB3 port AND external enclosure) has a limit of 0.5 amps power to the devices attached to it. So even if you are plugging ONLY this enclosure and HDD into one USB2 port, it may require more power than that to work.
There are many "Laptop Portable USB Hard Drives" on the market that have dealt with this issue in two ways. The simplest was to use a special cable from the enclosure to the computer that has TWO USB connectors on one end, and you must plug in both of them to get enough power (from two computer USB2 ports) to get it to work. In addition, most of these units contain a special design of low-power-use HDD that turns a a slower speed so that it can function with this limited power supply.
I don't know what old HDD you took out of your old laptop, and what power it requires. So I can't tell you whether it MIGHT work if you could get one of those special two-header USB connector cables and plug your system into two of your new laptop's USB2 ports. Maybe.
The other option is to get a different kind of external enclosure that will accept the 2½" old HDD and will use a USB2 cable to connect to your laptop, but ALSO has its own power supply module ("wall-wart" or "power brick") that supplies to the enclosure (and its HDD) all the power needed without relying on the USB2 port to supply it. In that direction there are two types of devices.
A popular type now is the "dock' that is just an open block you can plug your HDD into and connect to a computer. These are designed for easy insertion and removal of the HDD so you can use one dock for many HDD's. Here's an example:
http://www.newegg.com/Product/Product.aspx?Item=0VN-0003-000H5
It is designed for use with USB3 ports and comes with its own USB3 cable. However, the USB3 cable system is designed so you CAN plug that into a USB2 port on a computer and it will function properly at the slower USB2 speeds. This dock can handle 2½" and 3½" (desktop) HDD units and comes with its own power brick.
The second type is an enclosure designed more for "permanent" installation of one HDD in a case that you then close up. You CAN open it and change to another drive, but it's not intended for you to do that often. This enclosure type provides more protection for the HDD inside. An example:
http://www.amazon.com/Inateck-Aluminum-External-Enclosure-Optimized/dp/B00UAA4J6G/ref=sr_1_1?ie=UTF8&qid=1462073676&sr=8-1-spons&keywords=hard+drive+enclosure+3.5&psc=1
Click particularly on the enclosure/case model that sells for $24.99. This also is set for USB3 and comes with its power brick and USB3 cable, plus a small spacer pad useful with thin small laptop drives.
Before proceeding with either of these, check two things:
1. Does the enclosure you have already also have its own power supply module you can attach to it?
2. Just to be sure the HDD unit is not faulty, temporarily connect it to a desktop system as an internal drive, using a SATA power connection and a SATA data cable to a mobo port. Verify that the drive works before spending money on an external enclosure.
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