TRACON Controller's improper coordination and STARS equipment limitations contributed to a loss of separation event.

Narrative

[I was] working 3 positions combined. Weather [was] IFR with some [aircraft] deviating. [The] supervisor [was] getting currency on position. I requested a split and received no response from CIC. A second Controller requested to split and was told by CIC; 'They will be fine.' Aircraft X departed for ZZZ1 and was vectored around traffic and weather. When aircraft X was south of all known traffic he was level at 4;000 FT on 170 heading and handed off to and shipped to the adjacent Approach Control; as per letter. I observed Aircraft Y (at 3;000 FT) on E-tag and assumed he was inbound ZZZ2 as I had no strip and STARS shows no information on where or what local traffic is doing.I took care of some things on the northern part of my airspace and received a call; 'ZZZ Approach; aircraft Y out of 3;800 FT for 6;000 FT; 270 heading.' I looked down to see aircraft Y climbing 3.5 miles away from aircraft X on a perfect intercept; still on a another sector tag and still with out a strip. It appeared as the adjacent Approach Control was descending and turning aircraft X to the northeast to avoid a collision (They do not have control for descent; but it was an excellent choice). I turned aircraft Y south and expedited his climb. I believe we had to be within 3 miles. After I knew the aircraft were separated; I turned aircraft Y back west (290) and requested the hand off and strip.I came from an ARTCC; having worked there for 20 years. I would scrap STARS and bring in the displays and data presentation from the Center. Over 2/3 of the time in the TRACON you have no way of knowing what or where an aircraft is going or doing without a strip. This is unnecessarily dangerous with the overlapping and shared airspace. At the Center; anyone could tell [more aircraft flight plan data] just by looking at a data display.