Cell towers only transmit around 10 watts usually. Sometimes up to 50 or so in urban areas. Your phone can transmit up to 2 watts.
Transmit power does obviously have a big effect on the range of a signal, but it is not nearly as much as you would think due to the 1/r2 relationship of radio waves propagating out from the source. If your transmitter puts out 4 times as much power, you only get twice the range. 16 times the power, only 4 times the range. 100 times the power... 10 times the range.
The main advantage a cell tower has over your phone is in the size, power, complexity, and quality of the low-noise amplifier in its receiver. Since the tower is not limited so much by size and cost, they can put a really high quality amplifier in the tower (probably a noise figure of 1-2 dB...) and very effective low-loss filters in front of the amp (probably a cavity filter which is 10x the size and weight of a cell phone all by itself). This has a huge effect on the quality of the received signal compared to the cheap, small stuff that has to fit into the cell phone.
You might think that the tower's big antenna means it can "hear" small signals better compared to the phone's tiny antenna, but this is not true. The path from the cell tower to the phone is 100% symmetrical to the reverse path. In other words, when the tower transmits to the phone it gets a big benefit from its large antenna because the power is transmitted mostly in the direction of the phone, but this same benefit is in effect when the tower is receiving signals from the phone. The phone's tiny antenna has equally lousy performance when receiving signals from the tower as it does when it is transmitting back out to the tower.