FeedBurner is probably the best option to start with for most blogs. It's free and gets the basic job done. The only reason you might want to upgrade to FeedmailPro is the same reason people upgrade to Aweber: you might need some more advanced features.
Here are a few signs you might have outgrown FeedBurner:
If you want more flexible scheduling. The only option in FeedBurner is to send daily emails and this can overload your subscribers causing them to unsubscribe. With FeedmailPro you can schedule delivery to weekly or whatever you'd like.
If you want a "thank you" email to go out with a sign-up bonus or ebook after someone subscribes. Many bloggers use sign-up bonuses to increase readership and you can't do this with FeedBurner.
If you want to send out the occasional email blast to your list that isn’t a blog post. Some bloggers use this for affiliate promotions or to create special content only for people on their email list.
If you signed up for Aweber a while ago and are now locked in, paying higher fees every month, FeedmailPro offers you a way out. You can move your list without requiring another opt-in and losing half your subscribers.
If you want to automate sending the subscription email to users who register elsewhere on your site. Let's say your company has a registration page. You can put a checkbox there that says "also subscribe me to your blog" and it will automatically send them the confirmation email to subscribe to your blog. This advanced feature can drastically increase blog readership for many companies.
If you'd like to segment your list into different groups to send them different content or marketing.
To sum it up, for most blogs a simple and free solution like FeedBurner is a great start. If you're ready to start monetizing your blog or doing something more advanced, Aweber was what people always used in the past but it was very expensive. Now you can get the same advanced features at a much lower cost with FeedmailPro.