notar vs fijarse
Notar is to notice or sense passively. Fijarse is to pay attention on purpose — to look carefully.
notar
To notice, to sense, to feel that something is the case.
Noto que estás cansado.
I can tell you're tired.
Perception without effort.
Se nota que has estudiado.
It shows that you've studied.
Se nota = it's obvious, it shows. Very common.
No noto la diferencia.
I don't notice the difference.
Not perceiving something.
fijarse
To pay attention, to look closely, to notice on purpose — fijarse en.
Fíjate en este detalle.
Look closely at this detail.
Directing attention. Fijarse en + thing.
¿Te has fijado en su sonrisa?
Have you noticed her smile?
Asking if someone paid attention.
¡Fíjate!
Just imagine! / Would you believe it!
Idiomatic reaction to surprising news.
Tricky bit
'Noté su corbata' is just a passing observation. 'Me fijé en su corbata' means you paid attention to it, perhaps because it was odd or striking.