I'm thinking if you like telecasters, you're probably going to enjoy other fenders. A quick search will tell you the difference between neck "scales", fender tend to be shorter, gibson longer, PRS in between and most other companies are subs of either Fender or Gibson
You have it backwards... Fender scale is something like "25.5 and Gibson is "24.75. It's not a huge difference, but it is notable.
To the OP, any info about the make and model of your dad's Tele? I'm rather partial to them myself, and I'd just be curious to know. Anyway, I wouldn't restrict your search to just Fender/Gibson; a lot of people get really caught up in the brand name even if these big name companies often rest on their laurels and are sometimes outstripped by rivals or even their own subsidiary brands. Don't be afraid to look into knockoffs, this is especially true of Gibsons, but the catch there is that the high-quality knockoffs (Navigator/Edwards, Tokai, etc) can still be fairly pricey. I certainly would be worried about spending less than 600 on something that says "Gibson" on the headstock, so just be sure to read up. As far as Fenders go, a lot of people have given great praise to the Squier "Classic Vibe" models which apparently equal or better the Fender Mexican standards for under $400. Unfortunately though, I did play a couple in a shop and felt the neck finishes to be very sticky and plastic-feeling, but that's just one man's opinion.
Anyway, maybe it's a bit early for you to know, but the style of music you play should factor into what kind of guitar you want. To me, so much of it is about the neck. First of all, there's scale length, as was mentioned earlier in my response to Lord Elendiir. Gibsons are have a bit of a shorter neck, which can make things a little easier to play because you don't have to reach as far from fret to fret, and the strings will feel slightly spongier. However, the higher frets might feel cramped, or maybe you have larger hands, so there's that. On this issue I will say that I mostly play Fender scale necks, but I will admit that the smoothest playing guitar I ever played was a $5000 Navigator brand Les Paul knockoff in Japan. Besides the length of the neck, there's also the profile, which is the thickness. Vintage style necks will normally be a good bit fatter, and then you have the thinner necks which are generally more seen on "shred" guitars. Apart from the neck profile, there's the fingerboard radius. Fingerboard radius is expressed as the radius of a circle, so a neck with a smaller radius will have a rounder fingerboard, and a larger radius will have a flatter fingerboard. Many people automatically opt for the flatter radii in saying that it allows you to lower the action (how far you have to press the strings) and bend more easily without "fretting out" or causing any fret noise. This is particularly notable on my Japanese Fender 62' style Telecaster, which has the roundest radius you can get, "7.5. However, many people find rounder radii more comfortable for rhythm work, particularly barre chords. One compromise which you can find on some guitars (like my Charvel San Dimas and most Warmoth necks) is a compound radius which is rounder at the lower end and higher at the treble end so it's made a bit easier for both "rhythm" and "lead" styles. Another thing to consider is neck wood. The vast majority of necks are made out of maple, but fingerboards are commonly either rosewood or maple (if it's maple, then the neck/fingerboard can be just one contiguous piece), and sometimes ebony. It'll be hard to say which you prefer before you get some playing experience, and there's a little bit of a difference in sound, but maple feels a bit harder, and rosewood a bit softer and is said to collect/interact with the oils from your hands more noticeably. I don't have much experience with ebony, but I believe it's supposed to have the hard feel of maple with the mellower sound of ebony. However, as was mentioned earlier in this thread, wood has a much lower effect on the sound of an electric guitar than an acoustic. A last thing to consider with the neck is how many frets it has. Vintage style guitars tend around 21, but 22 is more common these days, and 24 is not uncommon. This just gives you access to a few higher notes. 24 fret necks are generally more commonly a feature on "shred" style guitars, though my Charvel only has 22.
Aside from neck issues, you'll want to consider what you might want out of pickups and tremolo units, but I don't want to go into the same amount of detail on these items as I have on necks and make this post an unreadable novel-length treatise. My advice to you is, play some guitars in local stores, then go home and do lots of internet research on the ones you liked. However, in my mind, playing in store isn't the holy grail of information gathering that some people make it out to be. Many stores don't actually keep their instruments that well set-up, so even a nice guitar can feel like complete crap. Both of my main guitars were purchased online and had better setups out of the box than many instruments I've played in stores. Don't be afraid to buy used!!! You can get great deals. My Charvel (which I keep mentioning because I'm still in the obsessed honeymoon phase) was a used mint which ran about half of what it would have new, so I in fact got an $1100 dollar guitar at your maximum price point of $600.