They are close in meaning; if you were to ask someone to quantify what percentage of similarity they mean, you might find that they were more or less the same. However, in rhetorical effect, they are quite different.
"Pretty much the same" means the two meanings might not be exactly the same, but the difference is negligible. They can be treated as if they were the same, for most relevant purposes in the speaker's opinion.
"Almost the same" emphasizes the similarity, but also emphasizes the small difference. If someone called two things "almost the same," you'd expect them to then explain what the difference is and why it's important, assuming it's not already clear from context.
This is not always true, but it tends to be.