What is the correct term used to describe this tense in English — Present Progressive or Present Continuous? I see both terms used in grammar books.
To get technical about it, present is a tense, but continuous and progressive are grammatical aspects, not tenses. Whereas tenses mark when an action happens, aspects give other temporal information, such as duration, completion, or frequency. English makes no distinction between continuous and progressive, and they are both formed using the present participle (–ing verb forms). Some languages, such as Chinese, distinguish between continuous and progressive, where the continuous aspect refers to the continuing state of the subject, whereas progressive refers to the the fact that the action is in progress.
The Wikipedia article linked above gives an example from Cantonese, where verb suffixes are used to mark continuous and progressive. The plain sentence I wear clothes would be translated as I am wearing clothes if the verb suffix for the continuous aspect were used, and as I am putting on clothes if the verb suffix for progressive aspect were used.
These are synonyms. Either meaning is understood.
Since I've been getting flak for this answer (after six years this answer has been on the board), I'll elaborate a little.
First, let's look at what the word synonym means. A synonym is
A word or phrase that means exactly or nearly the same as another word or phrase in the same language
Let us also stipulate that the OP asked about English, not all languages. It's in his first sentence.
And now we move to the distinction, or lack thereof in English.
In the grammars of many languages the two terms are used interchangeably. This is the case with English: a construction such as "He is washing" may be described either as present continuous or as present progressive.
From what I know, English does not differentiate between the progressive and continuous aspects.
Consider the following sentence: 'I am wearing a red hat today'. In a progressive nature, this could not be comprehended because it is usually impossible to perform the action of wearing a hat for let's say 8 hours progressively. Imagine someone trying to put on a hat from 6 a.m. and successfully completing it 8 hours later. But if we are to look at continuous nature, the sentence makes sense because 'wearing' in this context means I continuously wear the red hat throughout the day. I have finished putting it on at 6 am and continuously wear it for the rest of the day.