Which image compression is better, LZW or ZIP? I am using Lightroom to export images.
Better is a relative term and, to some degree, will vary in terms of amount between the two depending on a variety of factors including the bit-depth, frequency of discrete colours, etc. Some experimentation may be necessary on this front, though my reading indicates LZW is good for lower bit-depth images with lots of the same colours and tones in it and ZIP for when that is not the case. In other words, if the image is 8 bits go LZW and if it is 16 bits go with ZIP, as a rule of thumb, but with the caveat that it's not an absolute rule and there may be exceptions.
The only other thing I'd note is that LZW has been in the TIFF standard since 1992 and ZIP since 2002 (as part of a supplement when Adobe added it). While that's probably more than enough time for it to no longer be an issue, there may be the odd piece of software out there that handles LZW compression but not ZIP.
Compression is something you can see yourself, so I'll focus on interoperability and long-term preservation.
The EU's Succeed 2014 Recommendations for metadata and data formats for online availability and long-term preservation recommend "Uncompressed or LZW compression" for TIFF masters (p. 68) and note that «If files are actively managed in a digital repository, it is possible to consider using either LZW or ZIP lossless compression for the TIFF files. JPEG compression should not be used within the TIFF format. [...] Most of the respondents use uncompressed images (64%), if compression is used then LZW is mostly used».
In practice, I'm not sure there is a difference for single-page TIFF files. I did find problematic TIFF files whose compression upset my open-source program in the past, but I don't remember what was the exact culprit. LZW was patented until 2003. Based on the data above, however, it's possible that commercial support is more used to LZW and some software may still be poorly tested with ZIP/deflate TIFFs...
Make sure not to introduce some data loss yourself. It's easy to accidentally strip EXIF or IPTC/XMP metadata from your files while converting. Example commands with imagemagick and vips: