Blogger is an entry level blogging platform but can't be used for high means. Some reasons are as follows
1. It can't handle high band witch, chances are there your account will be suspended if you violate even one TOC.
2. Its a Google thing, as we all know recently Google deleted its Reader without letting anyone know this may happen with Blogger too and you will loose all of your hard work.
3. Those low-profile themes will not make you happy all the time because you will like to change them whenever you take a look to any another which will give your Blog a poor ranking. You will feel lack of something, whenever you will look at any Wordpress blog you will feel jealous. It happened with me too.
4. If anything occurs with your Blog like file corruption or anything else, then you are blown away you can't do anything, but with Wordpress nothing like this happens. Wordpress platform can be corrupt but it will not affect file saved in the hosting.
5. You will not get those awesome plugins on blogger and it will make you blogging journey really bad. You will feel that you are seriously tortured because you can't leave your hard work and can't work with it too.
Benefits of Self-Hosting
1. Access to core files, If anything happens you can just access them and repair the issue.
2. We have Wordpress which is best CMS out there, most of the famous blogs use Wordpress as there platform because of its awesome functionality.
3. There are many more things which Blogger can't do but it is work of some minutes for Wordpress.
If you want to take your blog to a next level then blindly go for Self-Hosting.
I am currently pursuing High School and planning to pursue career in Computer. Currently, I work on one of my blog - JTech Articles.