を兼(か)ねてAnd also, Also (partly) to, Also (partly) for (the purpose of), At the same time, To double as, When … also, While … as well, And as

Both are similar in that they describe doing something 'along with' or 'incidentally' to another action. かたがた is particularly formal and used especially in writing, e.g. お礼かたがたお伺いしたく存じます ('I hope to visit you, in part to thank you.'), while を兼ねて (and variant forms like も兼ねて) are still somewhat formal but used more widely, e.g. ダイエットも兼ねて毎朝散歩する 'to walk every morning, in part for purposes of a diet'.

を兼ねて and かねる come from the same verb, but their uses differ. を兼ねて expresses that something serves two purposes simultaneously, as in 運動を兼ねて散歩してきた (I went for a walk, also to get some exercise). かねる attaches to verb stems to express inability politely or indirectly, as in 私には分かりかねます (I’m afraid I don’t understand).