Disadvantages of both is that they are both tape based.
JVC GY-HD111E:
Pros:
2 XLR microphone inputs
Interchangeable lenses
Cons:
DV/HDV
Recording Formats:
720/24p, 720/25p, 720/30p, 576/50p, 480/60p, 480/24p, 480/60i
Though the ability to shoot in progressive instead of interlaced would be an advantage.
Source:
http/www.bhphotovideo.com/c/product/513194-DEMO/JVC_GYHD111E_GYHD111E_Professional_PAL_HDV.html
Sony HDR-FX1E
Pros:
1080i/60i but please note this is HDV (1440 x 1080) and will not be true HD (1920 x 1080) and is interlaced.
3x 16:9 1/3" Mega-Pixel CCD - 1,120,000 Pixels per CCD
Video Actual Pixels 1,070,000 per CCD
Cons:
No XLR microphone inputs
Lack on interchangeable lenses
Source:
http/www.bhphotovideo.com/c/product/356387-REG/Sony_HDRFX1E_HDR_FX1E_PAL_HDV_1080i.html
You might want to do a little more research on both to see how they compare. Depending on how much they are selling for there may be better options.