If it's a thicker TV, then its backlight is a CCFL - a fluorescent lamp. While these have a theoretical lifespan of tens of thousands of hours, they start to color shift after a few hundred hours, and dim after a few thousand. They also need in the neighborhood of a thousand volts to operate, so your TV also needs a step-up inverter inside just to drive the CCFL. This component ends up being extremely sensitive to power spikes and brownouts, and is another failure mode which effectively cuts down the lifespan of the backlight, and increases the cost. (While inverters themselves are cheap and you could theoretically replace just a burnt out inverter, getting the proper one for your TV or laptop is sometimes prohibitively expensive.)
The newer, thinner TVs use an LED backlight (sometimes called LED TVs). Those are much more reliable, simpler to drive, nearly immune to color shift, less prone to dim, consume less power, and are relatively inexpensive to replace if they die (unless your manufacturer did something nuts and incorporated the backlight into the panel itself).
If the old TV is working fine, these is little reason to replace it. The analog port inputs is a genuine advantage if you have older analog devices (newer TVs can't even accept analog signals of any kind - you'd need a separate analog to HDMI adapter). But you'll be burning about twice the power of a modern TV (albeit even that is still only around 50-100 Watts. And the picture is probably considerably dimmer than when the TV was new. So there are also good reasons to upgrade.