|450 - 300 = 150 watts is too much of a difference.|
That's output watts by the way. Computer power supplies are never 100% efficient - the best of them are no more than about 85% efficient.
The chances are that if you don't upgrade the power supply, the 300 watt power supply will be overloaded all the time when the 5750 is installed and, if the computer starts up at all with the card installed, it will eventually malfunction, and after a longer time fail completely.
When a power supply malfunctions, or especially when it fails completely, it can fry the mboard or anything connected to it.
If you want to take that chance instead of upgrading the power supply, which isn't expensive, go ahead.
In most if not all cases, the max capacity rating of the PS is an intermittent rating (unless it says continuos on the label). It's recommended that you do not load your PS to any more that 80% of that rating if you are going to be using something that puts a constant load on it, such as playing a recent game for hours on end. In that case, you multiply the min capacity stated for the system with the particular video chipset on the card by 1.25 to find the min. capacity of the PS you should get.
If you do upgrade the power supply...
Don't buy an el-cheapo (in quality) PS.
See response 3 in this: