On the face of it, using laptop drives in a server or NAS sounds strange. 2.5" drives are generally perceived as being anaemic, with poor performance. It is certainly true that a good 3.5" drive will outperform a good 2.5" one, but there is a lot more to it than that. The overall throughput of a server/NAS doesn't just depend upon the raw, theoretical speeds of the drives, but also derives from the performance of the controller, the design of other circuitry, the operating system, the network adapter and so on. By way of analogy - a car with a 4 litre engine may well be faster than one with a 1 litre engine, but it's likely to be 50% faster rather than 400% faster. In short, the performance of an individual 2.5" drive is not the limiting factor.
So, what is the case for 2.5" drives? Basically, they use less power (electricity) and are more reliable. The reliability stems directly from the reduced size and weight, as they vibrate less than their larger brethren (sisters?). Although vibration may seem minimal on modern disc drives, it is there and over time it has a significant physical impact on the drive: many of the drives that fail do so because they have literally been shaken to death. Again, because they are smaller and lighter, 2.5" drives use less power. Whilst this may be no more than a penny or two a day, consider a NAS that is on 24x7 for, say, 5 years and the savings are quite significant.
One downside with 2.5" drives is that they are less capacious than larger ones. 3.5" drives are currently available in capacities up to 4TB, whereas 2.5" ones max out at 2TB. That means that in a server or NAS with a given number of bays, you can only get half the capacity. And it will cost more, too: a quality 3.5" drive costs around £80 at the time of writing, whereas a 2.5" drive is £130.
Server and particularly NAS manufacturers are starting to cater for 2.5" drives. Whereas it is possible to use an adapter to enable the use of a 2.5" drive in any pretty much any NAS, some now come with drive bays that can hold either size (and it is even possible to mix the two sizes). An example of one supplier that is ahead of the curve is QNAP, most of whose models can use either. Arch-rival Synology even have a NAS - the DeskStation DS411slim - that is designed entirely around the 2.5" form factor. It may not fit in your pocket, but it's getting there!