What is the difference between

"ändern" and "verändern"

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"verändern" is more or less equivalent to "to alter" (it can also used reflexively though", while "ändern" denotes a more drastic type of change.

If you change your name (normally people pick a name that is quite different from their original one), you'd say "Ich habe meinen Namen geändert; ich habe meinen Namen ändern lassen.

If you hear

Mein Mann hat sich geändert.
, this denotes a significant change, "he's not the same person as before, maybe he is no longer violent or stopped drinking or something major like that)

On the other hand,

Du hast dich verändert.
sounds like you are denoting that the person has either altered their apperance (maybe dyed their hair), or maybe has grown up a bit.

This explanation covers at least most of the cases in my opinion, but there are some exceptions as well.