I wrote a text about our company's services and added that we also can offer products tailored to the needs of our clients. One of my colleagues insisted that the expression was wrong and I should say that we offered tailor-made products. Is there actually a difference between the two expressions generally or in the context I described?
There is a shade of difference between them.
Something that is tailor-made for someone was conceived, designed, and built to suit that specific customer.
Something that is tailored for someone could have started its existence as a fairly generic item, but it was most likely designed and constructed with the goal that at some point, somebody could come along and make alterations to it that would enable it to fit better onto a particular customer. (Many different possible alterations may be available, though, and in conflicting arrangements: A suit could be designed so that it can be either taken in at the waist, or let out, without looking like it has been modified.)
So if you're starting from scratch for your customer, then tailor-made is the better description; but if you've got a common base to start from, and you're making changes on top of that, then just tailored is more accurate.
A figurative meaning of the verb tailor is ‘to design or alter (something) to suit specific needs’ (OED), so the way you used it would seem appropriate. Equally, there is a verb, tailor-make meaning much the same thing.
l do not believe there is any difference in meaning between these two expressions although "tailored" may carry the additional meaning of "adapted to the needs of a specific group" to some. Using "bespoke" may avoid this confusion.