First off, there are really no standards - there are only recommendations. These recommendations are laid out by the World Wide Web Consortium (w3c)
A W3C Recommendation is a specification or set of guidelines that, after extensive consensus-building, has received the endorsement of W3C Members and the Director. W3C recommends the wide deployment of its Recommendations.
Web standards mainly concentrate on three areas:
Basically, what this means is that the code you have in your web page should be according to the definitions laid out in the document type you have chosen for your website. This makes it easier to identify potential errors, leads to faster web development and helps set up a better ground for maintenance.
Validation as a future-proof quality check Your website will be understood and parsed by newer browsers as they replace older ones.
Validation eases maintenance Creating Web pages or applications according to accepted coding standards makes them easier to maintain, even if the maintenance is done by someone other than the original developer.
Validation helps teach good practices Websites designed with validation in mind will always be nicely and neatly organised.
Validation is a sign of professionalism This is what will eventually separate professional designers from the snake oil salesemen.
The recommendations for good semantic code are simple. Use heading elements (<h1> to <h6>) for headings and sub headings, paragraph elements (<p>) for paragraphs, and lists (<ul>, <ol> etc) for lists and so on.
This will help you and search engines to more easily find certain parts in a web page and will dramatically decrease load time because of less code - it will also improve maintenance.
The main reasons for the separation are:
Reusability It is easy to reference code and re-use it across the entire web site.
Maintenance All the code is in one central location which makes applying all presentation and interaction code to all the web pages within a web site.
Creating valid, standards compliant websites is not a herculean task. It is very easily achievable and has almost instantaneous results. Even if a browser doesn't support a technology you use, the website will degrade gracefully and not deteriorate into a jumbled mess of html gibberish or end up looking like a cubism masterpiece. Which reminds me, you four people surfing this website on Internet Explorer, for the love of all that is moist and squishy, please switch to something else!