The main disadvantages of a private tour when climbing Kilimanjaro is that it works out to be more expensive for solo climbers or couples.
For single people it is a particularly expensive option as it means they will have to pay a single person supplement.
On a private tour it is also unlikely that you will get the same group bonding experience that you would on a group tour.
This can be a rather lonely experience if you are by yourself. However, you will still meet other hikers along the way and at campsites, so there is some opportunity to socialize with others.