Hi, this question was asked before. Check the answers here: https://www.researchgate.net/post/What_is_difference_between_TAE_and_TBE_buffers_and_their_properties_regarding_use_in_agarose_gel_electrophoresis
Besides what's been already posted, TAE tends to give sharper bands than TBE.
If you keep the stock of TAE at 50x, which is not possible for TBE, there will be no growth in it. As an added bonus, it occupies less room on your shelf! ;)
High voltage causes heating... a plausible reason for conversion of supercoiled to relaxed circle DNA.
TAE on the other hand is not very suitable for high voltage, sufficient heat can build up on cathode end for the gel to begin melting, which does wonders to resolution! ;)
When in a hurry (and aren't we all), pour a gel, put it in the gel box (mini set up) and stick the entire thing in a freezer for about 20 min while getting ready with sample preparation. Bringing the gel and buffer to near freezing gives one enough of an advantage to zap the mini gel at 150V (constant) to get usable results in about 10 min.
This used to work quite well with old Hoeffer boxes that had fillable compartment under the gel stage. I still have a couple of them in my lab - not sure if anyone sells them now. Filling the compartment with ethylene glycol or 50% Glycerol gave a little extra capacity to absorb the heat so gels could be "zapped" for 5 more min for better resolution.
"TAE is advantageous for high resolution of long nucleic acid fragments (longer than 1500 bp) on agarose gels. It has a lower buffering capacity than TBE and in general, nucleic acid fragments move slower in TAE gels (apart from linear dsDNA, which tends to run faster). TBE has a greater buffering capacity and will give sharper resolution than TAE. However, TBE gels in general afford a poor recovery of nucleic acids compared with TAE gels. TAE is also used for native (non-denaturing) RNA analysis and in denaturing gels (instead of MOPS buffer) using prior denaturation of the RNA samples in hot formamide."
My lab uses TAE for everything because we don't see much of a difference in practice. I resolve very similar sized bands in TAE just fine, though apparently TBE should be even better.
TBE is better than TAE if you have to run a gel at high voltage or over a long time (like in case you have huge bands >5kb), because it doesn't melt as easy as TAE.
But -according to an old edition of Sambrook & Maniatis- the boron ions in TBE can inhibit ligation in cloning reactions.