Good question! 'Currently' and 'at the present' mean the same thing, they both talk about what is happening right now. 'Currently' is much more common and sounds more natural in most situations. For example, you can say: 'I am currently working' or 'I am working at the present,' but the first one sounds more natural. Often people add the word "moment" to the end of "at the present" like this: "At the present moment, we do not have any availability" ..Hope that helps, Lucy!

both carries same meaning

I'd say "currently" or "at the present time". I've haven't heard of just saying, "at the present..." You can also say "at the moment". They all mean the same.

There's no difference in meaning but in my view 'currently' is more formal, usually used in writing, or a job interview, while 'at present' or 'at the moment' are usually used in everyday speaking. I note that Americans seem to prefer 'currently' but I never use it.

Grammar is the only big difference. According to basic common modern literal definition: Currently = Presently But there are etymological undertones which can affect things stylistically. 'Currently' comes from 'current' which refers to the strong flow of water, while 'present' can also mean 'show' or 'give' or 'gift'.