It looks like both are in regular use today, and that this is also the case for “payer”, “balayer”, as well as any other verb in “-ayer”. I am not aware of any rule mandating to use one rather than the other when writing vs. speaking.

The pronunciation is slightly different though: “j’essaye” would sound at the end like “pareil”, whereas “j’essaie” would sound like “sait”.

I am not aware that one pronunciation or the other would have this or that connotation. It is even possible for native speakers to use both in the same conversation.