Magento – Difference Between getChildHtml and getChildBlock

getBlockHtml('block_name_here') get's you the html of the block with name block_name_here if it finds it in the layout...anywhere.
getChildHtml('block_name_here') get's you the html of the block with name block_name_here only if that block is a child of the current block.

Here is an example. Consider the next layout section

<block type="core/template" template="some_template.phtml" name="some_name" />
<block type="core/template" template="some_other_template.phtml" name="some_other_name" />

Now in the template file some_template.phtml if you add this line of code <?php echo $this->getBlockHtml('some_other_name')?> you will get the html for the block some_other_name.
If you add <?php echo $this->getChildHtml('some_other_name')?> you will get nothing.

IN the following scenario

<block type="core/template" template="some_template.phtml" name="some_name">
    <block type="core/template" template="some_other_template.phtml" name="some_other_name" />
</block>

getChildHtml and getBlockHtml will get you the same thing. the html of the block some_other_name.

append appends an existing block that is referenced by name, to a new parent. It can be used to move a block to another location or to duplicate it.

<block> always creates a new block.