Question: My internship mission consists in recoding an entire software which was coded 8 years ago by a current senior developer of the company in another language.

The only trouble is that I changed the organization of the software from bottom to top. I switched to MVC instead of the "all in a file" pattern for example.

How do I show him my current work without sounding like "your previous work sucked so much, that's why mine is so different" ?

Answer: 
Was this person a senior developer when they write the code 8 years ago?  

Probably not - so the code wasn't 'senior developer quality'. 

Unless they are still writing code the same way they did back then, you should have little concern in showing how you've improved on their work using current best practices.
If it was my 8 year old code that you rewrote in a new language, while changing the structure completely: First, I know for a fact that the code that I wrote in 2009 should be rewritten in a different language if the company wants to continue developing it in the long term. And I know that it needs significant changes, because I knew in 2009 that parts of it could be expected to stop working soon. 

My question would be: How much responsibility am I expected to have towards that code? If you changed the language *and* the structure at the same time, that's a brave move. And you are on your own. You want to show me your work? Actually, I'm not interested. It's your code. If it works well, be proud of it. If it doesn't work, your problem. I'm not going to help you. 

What I'm saying is that you are much too late. You might have come to me before you rewrote it. I could have told you what parts of the code are good, what parts are bad because I never had time to do them well, and which parts are ugly because they contain sometimes bizarre workarounds that will make the app stop working if you "fix" the ugly bits. 

You worry about sounding like "your previous work sucked so much, that's why mine is so different". I don't know if that senior developer was an intern eight years ago, or was experienced back then. In the former case, he might think "your code is different, and it sucks just as much or even more". If he was senior back then, he might come up with a list of things you've broken.