If you want to preserve the size of your raster, you have to specify that in the gdalwarp command line.
Otherwise gdalwarp tries an own guess, based on the average of x and y source resolution preserving the extent, leading to square raster cells. It is pure coincidence that the pixel and line values are swapped in your case.
You can force to preserve the size with:
gdalwarp -overwrite -ts 466 427 grid_tif.tif warped.tif
gdal_translate -of AAIGRID warped.tif warped.asc
gdalinfo warped.asc >warpout.txt
and you get:
Driver: AAIGrid/Arc/Info ASCII Grid
Files: warped.asc
warped.asc.aux.xml
warped.prj
Size is 466, 427
Coordinate System is:
PROJCS["ETRS89_UTM_zone_32N",
GEOGCS["GCS_ETRS_1989",
DATUM["European_Terrestrial_Reference_System_1989",
SPHEROID["GRS_1980",6378137,298.257222101]],
PRIMEM["Greenwich",0],
UNIT["Degree",0.017453292519943295]],
PROJECTION["Transverse_Mercator"],
PARAMETER["latitude_of_origin",0],
PARAMETER["central_meridian",9],
PARAMETER["scale_factor",0.9996],
PARAMETER["false_easting",500000],
PARAMETER["false_northing",0],
UNIT["Meter",1]]
Origin = (-1181164.876702783400000,7834760.894102003400000)
Pixel Size = (6483.065191850767400,-7721.407482087573000)
Metadata:
AREA_OR_POINT=Area
Corner Coordinates:
Upper Left (-1181164.877, 7834760.894) ( 29d15'49.94"W, 65d45'28.20"N)
Lower Left (-1181164.877, 4537719.899) ( 10d26' 9.65"W, 39d19'37.84"N)
Upper Right ( 1839943.503, 7834760.894) ( 41d23'31.52"E, 67d22' 2.42"N)
Lower Right ( 1839943.503, 4537719.899) ( 24d38'45.72"E, 39d55' 6.12"N)
Center ( 329389.313, 6186240.397) ( 6d16'43.28"E, 55d47'29.86"N)
Band 1 Block=466x1 Type=Float32, ColorInterp=Gray
You can as well force to preserve the pixel resolution (but not both!):
gdalwarp -overwrite -tr 6486.620049172968720 7717.850831012333401 grid_tif.tif warped.tif
gdal_translate -of AAIGRID warped.tif warped.asc
gdalinfo warped.asc >warpout.txt
Which results in:
Driver: AAIGrid/Arc/Info ASCII Grid
Files: warped.asc
warped.asc.aux.xml
warped.prj
Size is 466, 427
Coordinate System is:
PROJCS["ETRS89_UTM_zone_32N",
GEOGCS["GCS_ETRS_1989",
DATUM["European_Terrestrial_Reference_System_1989",
SPHEROID["GRS_1980",6378137,298.257222101]],
PRIMEM["Greenwich",0],
UNIT["Degree",0.017453292519943295]],
PROJECTION["Transverse_Mercator"],
PARAMETER["latitude_of_origin",0],
PARAMETER["central_meridian",9],
PARAMETER["scale_factor",0.9996],
PARAMETER["false_easting",500000],
PARAMETER["false_northing",0],
UNIT["Meter",1]]
Origin = (-1181164.876702783400000,7834760.894102003400000)
Pixel Size = (6486.620049172968700,-7717.850831012333400)
Metadata:
AREA_OR_POINT=Area
Corner Coordinates:
Upper Left (-1181164.877, 7834760.894) ( 29d15'49.94"W, 65d45'28.20"N)
Lower Left (-1181164.877, 4539238.589) ( 10d26'23.03"W, 39d20'24.29"N)
Upper Right ( 1841600.066, 7834760.894) ( 41d25'26.15"E, 67d21'36.06"N)
Lower Right ( 1841600.066, 4539238.589) ( 24d40' 4.00"E, 39d55'44.24"N)
Center ( 330217.595, 6186999.742) ( 6d17'29.08"E, 55d47'55.44"N)
Band 1 Block=466x1 Type=Float32, ColorInterp=Gray
which has the same size, but a little expanded extent.
Direct gdal_translating your tif to asc is not possible, because the AAIGrid driver can not handle input with south-north ordering (although this is default with gdal_grid). See my answer on gdal_translate creates images that are mirrored for more details on that.
The gdalwarp output is ordered north-south and suitable for AAIGrid.
gdal_translateprints is "Producing a Golden Surfer style file with DX and DY instead of CELLSIZE since the input pixels are non-square. Use the FORCE_CELLSIZE=TRUE creation option to force use of DX for even though this will be distorted. Most ASCII Grid readers (ArcGIS included) do not support the DX and DY parameters." – user2856 Feb 05 '15 at 18:59