You mentioned log-in and that's what makes me think that you probably want to know the difference in the context of Internet. Well, sign up simply means to register. It could be portal, newsletter or things the like. So when you visit and access anything for the first time, you need to sign up. Often, this is referred to as register. For instance, if you are new to Twitter , you need to sign up first.
Having a quality UX is the make or break point for your app in the market. How good will an app be if its full of amazing functions but no one can use it effectively without tearing their hair apart in frustration? The sign-up screen of a mobile app is extremely important for conversion rates and its design is grueling. You need to consider efficiency and speed of the login process while maintaining simplicity, privacy, and data security. Keeping up with all these requirements on a simple screen is hard, but for those who manage to nail it, it gives better conversion rate and retention rates. From the user perspective, filling out signup forms is painful enough in websites, and put it in mobile apps, it is even more so.