The difference is that expression 1 looks for valid and full URIs, following the specification. So you get all full urls that are somewhere inside of the code. This is not really related to getting all links, because it doesn't match relative urls that are very often used, and it gets every url, not only the ones that are link targets.
The second looks for a tags and gets the content of the href attribute. So this one will get you every link. Except for one error* in that expression, it is quite safe to use it and it will work good enough to get you every link – it checks for enough differences that can appear, such as whitespace or other attributes.
*However there is one error in that expression, as it does not look for the closing quote of the href attribute, you should add that or you might match weird things:
/<a.*?href\s*=\s*["\']([^"\'>]+)["\'][^>]*>.*?<\/a>/si
edit in response to the comment:
To look for word inside of the link url, use:
/<a.*?href\s*=\s*["\']([^"\'>]*word[^"\'>]*)["\'][^>]*>.*?<\/a>/si
To look for word inside of the link text, use:
/<a.*?href\s*=\s*["\']([^"\'>]+)["\'][^>]*>.*?word.*?<\/a>/si