Κοντά σε κάτι - near something.
Δίπλα στην (for feminine nouns)/ στο (for neuter nouns) - next to something, usually right next to something.
I am next to you, which means I'm near you.
My house is near yours, but that doesn't mean it's (right) next to it.
That's why these words are not absolutely synonymous.
I hope the difference is clear ^.^
Mmm, not exactly. Someone could be close to you (κοντά σου), without necessarily being next to you (δίπλα σου). ^.^
I did answer that question in the comment right above. ^.^ It's because next to is not interchangeable with close to. Something could be close to you, without it necessarily being next to you.