As a Necro, I will try to answer this as un-biased as I can:
Both: Solo effectively. Probably the top solo classes in the game because of roots and pet tanking.
DPS: On shorter fights, I've seen Conj's do nice dps. On longer fighters Nec dots can steadily take a mob down. I think Conj's get more straight up nukes...where as Nec's are very limited in that area (Lifetap, Masterstrike, AA Dethreat, thats it).
Heals: Conj's do not heal. Necro's do. With AA our roots/stuns/etc can heal our pet without even using our life. Conj's get a nice stoneskin for their pet (Necro's can get a limited "absorb 3 hit" or "drain mana instead of damage" abilities with AA).
Pets: Both have 3 pet classes that are great for a variety of situations. Tank pets are pretty self-explanatory. Scout pets are nice for single target dps, and mage pets are nice for encounters (I also believe both class mage pets get some healing spells, at least I know necro mage pets do).
In the end I would say it comes down to preference and gear. A raid geared Conj will out DPS a solo content geared Nec, and vice versa.