Heja

Są to synonimy i często można je używać zamiennie.
'explain' jest mniej formalne niż 'clarify'

Pozdro

do tego co napisał quantum 137 można dodać że :
clarify  

Definition: explain, make clear
Synonyms: analyze, break down, clear up, define, delineate, draw a picture, elucidate, formulate, illuminate, illustrate, interpret, make perfectly clear, make plain, resolve, settle, shed light on, simplify, spell out, straighten out, throw light on
Antonyms: confuse, muddle

to jeszcze dodam parę przykładów użycia tych słów z OALD 8th Edition

CLARIFY

I am happy to clarify any points that are still unclear.
I hope I managed to clarify things a little.
Let me clarify my position on this matter.
The 1880s saw a major attempt to clarify the law and simplify court procedure. The illustrations are meant to clarify some of the points in the text.
There are one or two issues that need to be clarified.

EXPLAIN

I explained the issues in great detail.
I have a letter here which explains everything.
I know I'm late, but I can explain why.
It is a book which seeks to explain the birth of Greek tragedy.
It's difficult to explain exactly how the system works.
Let me explain what I mean.
Many theories purport to explain growth in terms of a single cause.
She explained the plan to me very carefully.
She saw his quick frown and hastened to explain.
She tried to explain about her fears and anxieties.
The doctor explained patiently what the treatment would be.
This phenomenon can be easily explained.
‘It was like this,’ she explained. ‘Let me explain,’ he continued.
Ed explained that his car had broken down.
First, let me explain the rules of the game.

po prostu zapamiętaj:

explain = wytłumaczyć coś w sposób bardziej zrozumiały
clarify = klarować
[np. sytuację: A situation on the stock market has been clarified. (sytuacja na giełdzie została wyklarowana), gdzie w tym kontekście brzmi lepiej niż:
A situation on the stock market has been explained (sytuacja na giełdzie została wyjasniona), które przyjmuje zupełnie inny sens znaczeniowy].