Your strength in a flow is your inborn ability to weave that flow with ease, and without getting tired. Egwene explained that she almost didn't have to think when weaving Air - because she is strong with it, whereas we see that Verin requires real effort to weave Earth when weaving a form of Compulsion on a fellow Aes Sedai.

Strength in the Power is the total amount of saidin or saidar which you can channel. It does not determine your strength in the 5 Powers directly, though I have always found that the stronger the channeler, the less likely they are to be weak in a flow. We have never seen a channeler of Egwene's strength or greater who has shown a deficiency in a flow. It may be assumed though - Egwene has true strength in Earth, and though Elayne and Nynaeve have never commented on weakness in the flow, we have to assume that they are in fact not as strong as Egwene in Earth. I've thus come to the conclusion that while strong channelers aren't necessarily strong in a flow, they can use raw strength to overcome their deficiencies. For instance, Nynaeve might only be strong in Air and Water, but she is so strong in saidar that she can weave huge amounts of Fire and Earth without tiring, and she can still appear to be strong in all five flows. On the other hand, channelers who are weak in saidar show true and severe deficiencies when they are weak in a flow. For instance, Elaida says that most Aes Sedai can't even fuse stone because they are so weak in Earth.

While using increasing strength in the Power does seem to correlate with increased strength in the Powers, it seems that artificially augmenting your strength with an angreal or sa'angreal still leaves you with the same weakness. Verin uses an angreal to weave Compulsion but Earth still hampers her to the same degree. RJ also confirmed that the leader's weakness is not obviated when linking.

Extremely strong channelers like the Forsaken are probably all strong in 4 or 5 Powers, meaning that they can weave the majority of weaves at a high level and for a long time without getting tired. Most average channelers probably struggle to weave in two or more flows and need to concentrate, and become fatigued after a while. Most weak channelers struggle to weave almost anything for a long time, and may actually not be able to do some things - like weave balefire, or fuse stone, or make cuendillar (which is partially a Talent too).

Hope this helps