First, never underestimate the budget Audtio Technica cartridges - they can be very good indeed - especially when fitted with upgraded styli.
Here is a selection for your 5625
http://www.turntableneedles.com/search.asp?keyword=5625&search=GO
Note that you can even get a HyperEliptical or Shibata stylus for it!
The Ortofon M20E cartridge is a better body than the 5625, but it is a series that is getting long in the tooth, and styli for it can be tricky - the better styli, when you can find them, tend to be quite expensive:
http://www.turntableneedles.com/search.asp?keyword=539-d&search=GO
The M20FL stylus is the Ortofon equivalent to the HE stylus for the 5625, at around 3x the price! - Yes it is better - not convinced about the value.
You can get an aftermarket "vividline" (HE equivalent) for around $80:
http://www.lpgear.com/product/ORTSM20FL.html
AT14S is another really nice cartridge, in its day it was close to top of the line from Audio Technica. - It is worth checking exactly what cartridge you have (and perhaps posting photos) as Audio Technica had the AT14/AT14Sa which had one type of stylus mounting and the AT14S which had a slightly different one - and they are NOT interchangeable!
The correct replacement stylus for an AT14S should be:
http://www.lpgear.com/product/ATSAT0014S.html
There was also a JVC branded version which was otherwise identical:
http://www.turntableneedles.com/Needle-612-DQ_p_1220.html
There is a range of styli that will fit:
http://www.turntableneedles.com/search.asp?keyword=201b&search=GO
So here is the deal - if budget is king, and $ are really tight - get a decent eliptical stylus for whichever one of these you choose - and get things playing music. - cost would be between $25 and $35.
A line contact stylus will sound better, last longer and reduce wear on the records... so I prefer to have stylus of that kind (HyperEliptical, Fineline, or Shibata).
The OEM styli are the best available, with the AT14S being the most reasonably priced of these.
The Aftermarket HE/Vividline styli at $80 have a great reputation, and you can get these for the Ortofon or the AT5625.
One final thing - the arm on your turntable is a mid-mass arm, so ideally you want a cartridge that is mid compliance - the best indicator of that is the VTF, and you should be aiming for a cartridge with a nominal (ie recommended running VTF not min or max) VTF of around 2g.
2g is the max for the Ortofon or the AT14s... so running these will tend to require more attention to turntable isolation, mounting, rack, etc... - as the resonance will be lower, and may make them sensitive to outside vibrations.
The AT5625 with a max VTF of 2.5g is the perfect match for your arm... and the easiest to use as a result!
Another factor to consider:
Different cartridges are designed for different loadings - the AT's usually work best with less than 200pf total capacitance. (Capacitance of the phono stage + capacitance of the turntable arm and interconnect cables)
The ortofon's are usually happy running with 400pf capacitance (which is much more common - and likely to be closer to what you have)
Check your phono stage capacitance load (should be in the spec sheet, or if you can find the circuit diagrams you can usually see the loading capacitors quite clearly) - if it is 50pf or lower, the AT's should be fine, if it is 150pf or over, then the Ortofon will be happiest with it.
Choose the body based on capitance matching & arm matching, then purchase the best stylus you can afford for it!
bye for now
David