Anfassen - Anfühlen - Berühren
all mean to touch by hand, but is there any difference between them?
"anfassen" implies conscious action and the focus is on the thing you touch.
"Fass das mal an! Fühlst Du wie weich das ist?" ("Touch that! Do you feel how soft it is?") "Ich mag nicht am Hals angefasst werden." ("I don't like being touched at the neck.")
"anfühlen" does not really imply "touch by hand". You touch something or are touched by something and the focus is on the sensation ("fühlen" = "to feel") the touch creates. Also you can not say "Ich fühle an", but only "Es fühlt sich an" , for example "Es fühlt sich weich an." ("It feels soft.") As another example, when you get a massage or when a doctor presses on your belly, you might be asked "Wie fühlt sich das an?" ("How does that feel?") It doesn't even imply physical contact, for example you might say "Das fühlt sich so gut an." ("That feels so good.") when you can finally sit down and relax after some hard work.
"berühren" is very neutral. It just describes the plain fact that you touch something either intentionally or accidentally.
I find this site pretty useful
Hope you like it, and a native german speaker will answer you in greater details.
Frohes neues Jahr :)