In terms of analysis of DNA by agarose gels made with 1 x TBA or 1 x TAE both effectively behave the same
TBE buffer is generally more stable than TAE and so can be re used multiple times; unlike TAE which breaks down by electrolysis more readily than TBE and so unlike TBE should only be used 1 x or 2 x ( I routinely use 1 x TBE up to 5 x)
Historically, if extracting DNA bands from gels TAE was considered better
Thus, if you are simply analysing bands on gels TBE is best
For gel extraction and DNA purification agarose gels made with 1 x TAE might be better although with modern extractions kits you can actually perform extraction from agarose made with TBE
I am not exactly sure what your DNA lengths might be, but I think TAE will work better for your genomic DNA although it wouldn't be wrong to use TBE. TAE gives a better and faster resolution of larger (about 2 kb or larger) DNA fragments, while TBE is best if using smaller DNA fragments of usually about 300 bp or less. In between these fragment sizes, though, either buffer should be applicable.
But also remember that the percentage or concentration of agarose gel you are using will also play a part. So 1% agarsoe gel for example will facilitate separation of very large fragments compared to using 2% agarose gel.
However I will add to and qualify that point by saying that percentage agarose will have a much bigger impact on resolution of large versus small DNA amplicons than whether you use TAE or TBE: specifically for both buffers a 2-3% gel will lead to better resolution for bands of 100-200bp and is better up to 500bp. Above 500bp 1% is better and for very large fragments >10kb 0.7% is better still. I also take Ismailas point that for very large fragments these 0.7% geos are best made with TBE. Nevertheless for fragments from 500bp to 10kb both buffers behave the same