Compared
SOP vs personal statement vs motivation letter

The short answer
A statement of purpose is programme-led, a personal statement is subject-led, and a motivation letter is a formal one-page business letter. The US and Canada ask for an SOP of 500 to 1,000 words; the UK and Ireland ask for a personal statement; German-speaking programmes ask for a Motivationsschreiben of 400 to 600 words. Sending the wrong one is the most common avoidable error.
Side by side
Statement of purpose: US, Canada, Australia, New Zealand. 500 to 1,000 words. Programme-led: what you want to study, why this department, what you will do with it. Faculty fit carries real weight for thesis programmes.
Personal statement: UK and Ireland. Postgraduate versions usually 500 to 1,000 words; UCAS undergraduate applications cap at 4,000 characters. Subject-led: why this subject, evidenced by what you have read and done beyond the syllabus. Around four fifths of it should be academic.
Motivationsschreiben: Germany and much of continental Europe. One page, 400 to 600 words, laid out as a formal letter with addresses and a date. Closer to module-level academic fit than to narrative.
What actually changes between them
Not the facts about you; those are the same in all three. What changes is the argument they are arranged into, and the register. An SOP argues forward from a research direction; a personal statement argues from demonstrated engagement with a subject; a motivation letter states fit briefly and formally.
The practical consequence is that converting one into another is not a matter of adjusting length. A UK personal statement cut to 600 words and given a German letterhead is still a UK personal statement, and it reads like one.
Letter of intent, and other names for the same thing
Some Canadian and Irish institutions and a number of professional programmes ask for a letter of intent. In most cases it is a statement of purpose under another name, usually shorter and more direct. Where both appear in one application, write to the word limits and the questions given rather than to the label.
Related questions
Can I use my SOP as a personal statement for a UK university?
Not without rewriting it. A UK personal statement is subject-led and expects roughly four fifths academic content: what you have read and done beyond the syllabus. An SOP reworked to fit the character count still argues in the US register, and UK admissions readers notice.