Since this was solved for non-distinct digits, I'm trying to get the maximum out of "MATH" with distinct digits.  
So the first try is with  

>! M = 9. We imediately see that S needs to be 8.
>! We get  
>! <pre>
>!  AWE + 
>! 8O9E = 
>! 9ATH
>! </pre>  

>! Since the hundreds of "AWE" and "MATH" need to be the same ("A") ?this means that "O" needs to be 0 and 9+W must not go over 10 so we don't get a 1 to carry. this means that W is also 0. So this does not work. 

So my seconds try is with  

>! M = 8 so S = 7  
>! <pre>
>!  AWE + 
>! 7O8E = 
>! 8ATH
>! </pre>  

>! Now we try to maximize "A". Let's go with the obvious 9. This means again O = 0 or 1 and we get.  
>! <pre>
>!  9WE + 
>! 708E = 
>! 89TH
>! </pre>  

>! now 8+w must not go over 10 so w = 1 ('cuz 0 is taken). But this leads to T being either 9 or 0 (if E+E go over 10) but both 9 and 0 are taken.  
>! continuing to maximize A.... so 9 does not work, 8 and 7 are taken by M and S....so 6   
>! <pre>
>!  6WE + 
>! 7O8E = 
>! 86TH
>! </pre>  

>! if O = 0 we end up with a similar contradiction as above, so lets try with o = 9 and w+8+(a possible carry from E+E) go over 10.  
>! <pre>
>!  6WE + 
>! 798E = 
>! 86TH
>! </pre>  

>! Now lets maximize T in math. 9, 8, 7, 6 are taken so lets go with 5.  This  means W+8+(possible carry from E+E) = 15. This leads to W being either 6 or 7 but they are already taken by A and S.  
>! so lets go with T = 4. This means W+8+(possible carry from E+E) = 14. So W is 5 or 6. 6 is taken by A so W = 5. 
>! <pre>
>!  65E + 
>! 798E = 
>! 864H
>! </pre>  

>! NOw E+ E needs to go over 10. but all of 5, 6, 7, 8, 9 are taken.  So this is no good.  
>! so let's try with T = 3.  This means W+8+(possible carry from E+E) = 13. So W is 4 or 5.
>! If w = 5
>! <pre>
>!  65E + 
>! 798E = 
>! 863H
>! </pre> 

>! E+E should be below 10. E cannot be 0 because H would be 0. SO it can be 1, 2, 4 (3 is taken by T). If it's 4 then H would be 8 which is taken by M. 
>! It works with both 1 and 2 but to maximize "math" we go with 2.  

So the final result: 

>! <pre>
>!  652 + 
>! 7982 = 
>! 8634
>! </pre>