The main problem with password-protecting a PDF file with a password is that you are basing the security on a password, which is some piece of data that a human user, somewhere, came up with in his mind, and was arrogant enough to deem "unguessable". It turns out that most passwords are guessable. The situation can be somewhat improved by making the password-to-key transformation expensive (this is called password hashing) but a weak password is still weak.
A second problem is that there is not one format for PDF encryption, but several. PDF encryption has a long history of custom schemes, the first of which taking root at times when the USA had strong, strict export rules for cryptographic-aware software; to make the story short, to allow the software to be exported without any administrative hassle, the crypto had to be laughably weak. Hence the encryption format described by @begueradj in his answer: the password is hashed, and only the first 40 bits of the results are kept as "master key" for the whole file. A 40-bit key is highly amenable to exhaustive search with today's computers, making the whole encryption thing a joke. It is now possible to make strongly-encrypted PDF files, that modern versions of Adobe Reader can process (I personally wrote some code to make PDF files that could be decrypted only with a smart card), but you have to do it explicitly.
Compounding the situation is the PDF internal structure. A PDF is a set of "objects", some of them being streams of other objects, or raw data. The whole idea is that the document should be amenable to a variety of accesses, e.g. jumping to any page within the document (possibly before having downloaded it whole), or extracting a table of contents. Since encryption is applied on a per-stream basis, the usual conclusion is that a lot of the document structure can be obtained without breaking the encryption (e.g. number of pages, length of each paragraph, number, size and position of pictures...). Whether this is a serious problem or not depends on the context, in particular why you want to encrypt. The real issue here is that the decision about what to encrypt and what not to encrypt is taken by some generic software that cannot, by definition, be aware of the context.
In practice, the point of password-protecting a PDF file is not to make it really inscrutable by eavesdropper; it is to document, in a clear and unavoidable way, that the file contents are sensitive and the file shall be handled with care. It is the equivalent of a red "top secret" stamp.