Ακολουθώ means follow (anything, it has a rather general meaning), while παρακολουθώ is usually used to describe that you are following somebody to see where he/she is going, with who he/she is speaking, etc. But there are many expressions where the verb παρακολουθώ can be translated in many different ways. Let's see the difference through some examples:
Ο αστυνομικός παρακολουθεί τον ύποπτο. = The policeman follows the suspect person. (he is walking behind him, he sees where he's going, etc)
Παρακολουθώ μαθήματα ελληνικών σε ένα ινστιτούτο. = I attend a Greek course at an institute.
Παρακολουθώ μία σειρά στην τηλεόραση. = I watch a series on TV. (I watch it regularly)
Χθες παρακολούθησα μία ενδιαφέρουσα διάλεξη στο πανεπιστήμιο. = Yesterday I attended an interesting lecture at the university.
Παρακολουθώ τις εξελίξεις στην Ελλάδα. = I read/watch the actuality in Greece.
Κάποιος με ακολουθεί. = Someone follows me. (he is just walking behind me)
Ακολουθώ τις οδηγίες βήμα προς βήμα. = I follow the guides step by step.
Τι ακολουθεί μετά το ρήμα στην πρόταση; = What follows after the verb in the sentence?
I'm sorry for the previous comment, I didn't have enough time to explain the difference well when I was writing it. :-)
An easy way to differentiate between ακολουθώ-παρακολουθώ is that ακολουθώ implies following with motion and παρακολουθώ implies following with vision (and/or hearing).
follow someone (by walking behind them) or follow a line (by walking on it)
follow (come right after, eg. in a tv program, or in a word sentence)
follow instructions (by doing them)
follow an example (by imitating - eg. Ακολουθώ τη μόδα (fashion))
watch tv, watch someone do something
stake out, follow with my eyes
monitor with hearing (Οι τηλεφωνικές γραμμές παρακολουθούνται (passive voice) = Phone lines are being watched)
attend a class/course
follow someone's speech, try to understand by listening carefully
follow/watch the news - eg. Παρακολουθώ τη μόδα (I follow and know what's in fashion but I may or may not imitate it; in contrast to "ακολουθώ τη μόδα" which means that I do imitate and wear what's in fashion. )