The GT 240 will be much faster, about 5 times as fast as 9400 GT, and it is currently the fastest card that does not require an auxiliary 6-pin PCI-e connector, but I don't think it is rated at 300W.
As wa1 asked, what is your PSU and what is its wattage?
Neither since one is crap while the other is overpriced. AVOID the low power cards since people have had many issues with these cards. A 4650/70 are good for the price and can easily beat the gt240. If you want nvidia get a 55nm 9600gt/9800gt that has the connector and down worry about load power it is quite low at stock clocks.
I really think a low power 9800 gt is fine (and cheap). I haven't heard of any specific issues except they are a hair less powerful than the regular versions. None of the connector-less ones have worse ratings, otherwise check and see if you have the necessary 6 pin (or 4 pin) PCIe connector or 2 extra molex connectors for an adapter.
I agree with the other posters. The GT240 is overpriced at the moment. Go with an ATI 4670. Your 300 watt power supply should handle it fine. There are low power 9600GT and 9800GT models that might also work and would be somewhat more powerful. However, the 4670 is easier to find and might be less likely to have problems. Stay away from the 4650 unless you can find a DDR3 model. I had a 4650 DDR2 and was quite disappointed in the performance.
Wow is it that difficult to chose a card :0 Just get what is in both budgets and is best suited to your needs. Most psu are over rated while a few slip through that can be overloaded by 5~10% safely for a few months. The GT200 cards are simply over priced for the performance they deliver. The 4550 and the 9400gt are priced a little better but I don't pay retail any more and for the same price you can lan better on ebay. I bought my second 9800gt 1gb for less than $90 and sli ever since. My 8800gtx that I am using in another comp only cost me $32 while I have seen other good cards for a little less.
A 4770 will most likely outperform the GT 240 (as it is compared to an HD 4650/4670 in Tom's review using $119 versions) for less money. An HD 4850 will destroy it if you can find it $100-110 these days. If you have to get NVidia, an 8800GT or 9800GT will also outperform it horribly (in a good way) for less money.
i'm in the same boat i got a dx4300-19 pc w/ a 300 watts, but w/ everthing eles i need to have a good up to date (for me!) all but the video card and power supply. i still plan on upgrading to a 650power
EVGA GeForce GTX 260 or maybe a 9800gtx+ which i had used this card and loved it! i should get the very best possible within reason
(any info on which 1 is welcomed good luck