A site upgrade involves replacing underlying software that is old and unsupported and with modern supported equivalents and can also be about modernising your website so that it presents better on the current broad scope of desktop and mobile devices. In an upgrade you should expect to see modern design techniques, mobile responsive displays, improved SEO and reduced risk of malicious infection and follow up service.
In terms of costs the older and more unmaintainable the original website is then the more the effort and costs will rise while with really old websites it is best to just start again, manually bringing forward the content to a brand new site for less than a migration might cost. I will always put forward what I believe to be the most cost effective approach when estimating the scope of an upgrade project.
Out dated websites are at higher risk of malicious attack, being discriminated against in Search Engines like Google and turning customers away because they can't view your site easily on their preferred device.