As an aviation photographer I often write down the registration to confirm the type at a later date particularly if a unusual design (ie at Oshkosh) or if uncertain of exact type.
Having said that some places are not keen on seeing people write down reg's but are quite happy fo them to take photos - had this happen at a show last year and could not work out the reasoning behind it at all. (so just took a photo making sure the reg was visible).
Aircraft during the course of their 'lives' may change registrations several times and it is always of interest (to me) to trace their history. At my local there is an AA5 which on doing a bit of research was one that I flew in the mid 80's when doing my ppl.