Especially
It is used to say that something applies more to one thing than to others.
That bar is always packed especially at weekends
Before adjectives
The underground was especially busy this morning
Specially
It is used to say that something is done for a particular purpose or a particular person.
These scissors are specially designed for left-handed people.
I made this cake specially for you.
With a particular way
It’s Halloween, so I rented this horror film specially for the occasion.
BOTH
I bought this milk especially / specially for you (especially is much more common in Br.E.)
With an adjective, they can both mean ‘more than normal’ but especially is much more common. In informal English, you can use that.
This vocabulary isn’t especially / specially / that difficult.
At the start of a sentence use in particular.
I love watching films. In particular, I really like horror films.
I love watching films, especially/particularly horror films.