Wood Forget me not and the one grown in many gardens are the same thing. So if you find some Forget me nots growing in the woods they may well be derived from plants that were once growing in a nearby garden.
They can be described as short lived perennials ( which seems a bit of a contradiction in terms) or as biennials which will survive more than two years. Either way I think you can see what I mean.
As you can see they are bright blue with a yellow ‘eye’. They flower in April and May. You can get white or even pink versions, no doubt from recent garden ‘escapes’
There are other species of Forget me not, in fact there are about 60 species native to Europe. I have one growing in my garden and recently at the entrance to our wood. It is Myosotis arvensis or field forget-me-not. The flowers are much smaller and light blue. Also the seed capsule sticks to your clothes and no doubt that is how it found its way into our woods. This one seems to die off after flowering so is more of an annual, though I have also seen reference to it being a ‘short lived perennial’
Forget me not’s are the food plant of a moth called the Setaceous Hebrew character, it is a brown moth and has a white mark on its fore wing which evidently looks something like a letter in the Hebrew alphabet. Here is a link to wikipedia where you can read all about it. Also the photo is one I have copied from that page.
There is a story about how they got their name which involves a Knight in shinning armor trying to woo a maiden with a bunch of blue flowers, but it all seems a bit far fetched. You may like to check out my page about the origin of common names of some wild flowers, including Myosotis sylvatica.