A: 

Did you check the spec?

Button:

<!ELEMENT BUTTON - -
     (%flow;)* -(A|%formctrl;|FORM|FIELDSET)
     -- push button -->
<!ATTLIST BUTTON
  %attrs;                              -- %coreattrs, %i18n, %events --
  name        CDATA          #IMPLIED
  value       CDATA          #IMPLIED  -- sent to server when submitted --
  type        (button|submit|reset) submit -- for use as form button --
  disabled    (disabled)     #IMPLIED  -- unavailable in this context --
  tabindex    NUMBER         #IMPLIED  -- position in tabbing order --
  accesskey   %Character;    #IMPLIED  -- accessibility key character --
  onfocus     %Script;       #IMPLIED  -- the element got the focus --
  onblur      %Script;       #IMPLIED  -- the element lost the focus --
  >

Input:

<!ENTITY % InputType
  "(TEXT | PASSWORD | CHECKBOX |
    RADIO | SUBMIT | RESET |
    FILE | HIDDEN | IMAGE | BUTTON)"
   >

<!-- attribute name required for all but submit and reset -->
<!ELEMENT INPUT - O EMPTY              -- form control -->
<!ATTLIST INPUT
  %attrs;                              -- %coreattrs, %i18n, %events --
  type        %InputType;    TEXT      -- what kind of widget is needed --
  name        CDATA          #IMPLIED  -- submit as part of form --
  value       CDATA          #IMPLIED  -- Specify for radio buttons and checkboxes --
  checked     (checked)      #IMPLIED  -- for radio buttons and check boxes --
  disabled    (disabled)     #IMPLIED  -- unavailable in this context --
  readonly    (readonly)     #IMPLIED  -- for text and passwd --
  size        CDATA          #IMPLIED  -- specific to each type of field --
  maxlength   NUMBER         #IMPLIED  -- max chars for text fields --
  src         %URI;          #IMPLIED  -- for fields with images --
  alt         CDATA          #IMPLIED  -- short description --
  usemap      %URI;          #IMPLIED  -- use client-side image map --
  ismap       (ismap)        #IMPLIED  -- use server-side image map --
  tabindex    NUMBER         #IMPLIED  -- position in tabbing order --
  accesskey   %Character;    #IMPLIED  -- accessibility key character --
  onfocus     %Script;       #IMPLIED  -- the element got the focus --
  onblur      %Script;       #IMPLIED  -- the element lost the focus --
  onselect    %Script;       #IMPLIED  -- some text was selected --
  onchange    %Script;       #IMPLIED  -- the element value was changed --
  accept      %ContentTypes; #IMPLIED  -- list of MIME types for file upload --
  >
Jonathan Sampson
+2  A: 

The difference is that <button> element can contain other elements like images, etc.

Ivan Zlatanov
+3  A: 

You should look at - button-vs-input-typebutton-which-to-use

adatapost
+2  A: 

The functionality is same. But the <button> tag has better rendering possibilities. As various browsers render the page elements in different styles, the usage of <button> tag gives an edge over input type is you want to make them look better and consistent over many browsers. You can also employ images in button tags.

For a better idea read http://particletree.com/features/rediscovering-the-button-element/

Christy John